assignment domain and range

Module 5: Function Basics

Domain and range of functions, learning outcomes.

• Find the domain of a function defined by an equation.
• Write the domain and range using standard notations.
• Find domain and range from a graph.
• Give domain and range of Toolkit Functions.
• Graph piecewise-defined functions.

If you’re in the mood for a scary movie, you may want to check out one of the five most popular horror movies of all time— I am Legend , Hannibal , The Ring , The Grudge , and The Conjuring . Figure 1 shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section we will investigate methods for determining the domain and range of functions such as these.

Figure 1. Based on data compiled by www.the-numbers.com .

Standard Notation for Defining Sets

There are several ways to define sets of numbers or mathematical objects. The reason we are introducing this here is because we often need to define the sets of numbers that make up the inputs and outputs of a function.

How we write sets that make up the domain and range of functions often depends on how the relation or function are defined or presented to us.  For example, we can use lists to describe the domain of functions that are given as sets of ordered pairs. If we are given an equation or graph, we might use inequalities or intervals to describe domain and range.

In this section, we will introduce the standard notation used to define sets, and give you a chance to practice writing sets in three ways, inequality notation, set-builder notation, and interval notation.

Consider the set [latex]\left\{x|10\le x<30\right\}[/latex], which describes the behavior of [latex]x[/latex] in set-builder notation. The braces [latex]\{\}[/latex] are read as “the set of,” and the vertical bar [latex]|[/latex] is read as “such that,” so we would read [latex]\left\{x|10\le x<30\right\}[/latex] as “the set of x -values such that 10 is less than or equal to [latex]x[/latex], and [latex]x[/latex] is less than 30.”

The table below compares inequality notation, set-builder notation, and interval notation.

To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol, [latex]\cup [/latex], to combine two unconnected intervals. For example, the union of the sets [latex]\left\{2,3,5\right\}[/latex] and [latex]\left\{4,6\right\}[/latex] is the set [latex]\left\{2,3,4,5,6\right\}[/latex]. It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

[latex]\left\{x|\text{ }|x|\ge 3\right\}=\left(-\infty ,-3\right]\cup \left[3,\infty \right)[/latex]

This video describes how to use interval notation to describe a set.

This video describes how to use Set-Builder notation to describe a set.

A General Note: Set-Builder Notation and Interval Notation

Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form [latex]\left\{x|\text{statement about }x\right\}[/latex] which is read as, “the set of all [latex]x[/latex] such that the statement about [latex]x[/latex] is true.” For example,

[latex]\left\{x|4<x\le 12\right\}[/latex]

Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,

[latex]\left(4,12\right][/latex]

How To: Given a line graph, describe the set of values using interval notation.

• Identify the intervals to be included in the set by determining where the heavy line overlays the real line.
• At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).
• At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).
• Use the union symbol [latex]\cup [/latex] to combine all intervals into one set.

Example: Describing Sets on the Real-Number Line

Describe the intervals of values shown below using inequality notation, set-builder notation, and interval notation.

To describe the values, [latex]x[/latex], included in the intervals shown, we would say, ” [latex]x[/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.”

Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.

Given the graph below, specify the graphed set in

• set-builder notation
• interval notation

Words: values that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3.

Set-builder notation: [latex]\left\{x|x\le -2\hspace{2mm}\text{or}\hspace{2mm}-1\le x<3\right\}[/latex];

Interval notation: [latex]\left(-\infty ,-2\right]\cup \left[-1,3\right)[/latex]

Write Domain and Range Given an Equation

In Functions and Function Notation, we were introduced to the concepts of domain and range . In this section we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.

We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products.

We can write the domain and range in interval notation , which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write [latex]\left(0,\text{ }100\right][/latex]. We will discuss interval notation in greater detail later.

Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.

Before we begin, let us review the conventions of interval notation:

• The smallest term from the interval is written first.
• The largest term in the interval is written second, following a comma.
• Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.
• Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.

Example: Finding the Domain of a Function as a Set of Ordered Pairs

Find the domain of the following function: [latex]\left\{\left(2,\text{ }10\right),\left(3,\text{ }10\right),\left(4,\text{ }20\right),\left(5,\text{ }30\right),\left(6,\text{ }40\right)\right\}[/latex] .

First identify the input values. The input value is the first coordinate in an ordered pair . There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.

[latex]\left\{2,3,4,5,6\right\}[/latex]

Find the domain of the function:

[latex]\left\{\left(-5,4\right),\left(0,0\right),\left(5,-4\right),\left(10,-8\right),\left(15,-12\right)\right\}[/latex]

[latex]\left\{-5,0,5,10,15\right\}[/latex]

How To: Given a function written in equation form, find the domain.

• Identify the input values.
• Identify any restrictions on the input and exclude those values from the domain.
• Write the domain in interval form, if possible.

Example: Finding the Domain of a Function

Find the domain of the function [latex]f\left(x\right)={x}^{2}-1[/latex].

The input value, shown by the variable [latex]x[/latex] in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.

In interval form the domain of [latex]f[/latex] is [latex]\left(-\infty ,\infty \right)[/latex].

Find the domain of the function: [latex]f\left(x\right)=5-x+{x}^{3}[/latex].

[latex]\left(-\infty ,\infty \right)[/latex]

How To: Given a function written in an equation form that includes a fraction, find the domain.

• Identify any restrictions on the input. If there is a denominator in the function’s formula, set the denominator equal to zero and solve for [latex]x[/latex] . These are the values that cannot be inputs in the function.
• Write the domain in interval form, making sure to exclude any restricted values from the domain.

Example: Finding the Domain of a Function Involving a Denominator (Rational Function)

Find the domain of the function [latex]f\left(x\right)=\dfrac{x+1}{2-x}[/latex].

When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for [latex]x[/latex].

[latex]\begin{align}2-x&=0 \\ -x&=-2 \\ x&=2 \end{align}[/latex]

Now, we will exclude 2 from the domain. The answers are all real numbers where [latex]x<2[/latex] or [latex]x>2[/latex]. We can use a symbol known as the union, [latex]\cup [/latex], to combine the two sets. In interval notation, we write the solution: [latex]\left(\mathrm{-\infty },2\right)\cup \left(2,\infty \right)[/latex].

In interval form, the domain of [latex]f[/latex] is [latex]\left(-\infty ,2\right)\cup \left(2,\infty \right)[/latex].

Find the domain of the function: [latex]f\left(x\right)=\dfrac{1+4x}{2x - 1}[/latex].

[latex]\left(-\infty ,\frac{1}{2}\right)\cup \left(\frac{1}{2},\infty \right)[/latex]

How To: Given a function written in equation form including an even root, find the domain.

• Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for [latex]x[/latex].
• The solution(s) are the domain of the function. If possible, write the answer in interval form.

Example: Finding the Domain of a Function with an Even Root

Find the domain of the function [latex]f\left(x\right)=\sqrt{7-x}[/latex].

When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.

Set the radicand greater than or equal to zero and solve for [latex]x[/latex].

[latex]\begin{align}7-x&\ge 0 \\ -x&\ge -7 \\ x&\le 7 \end{align}[/latex]

Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to [latex]7[/latex], or [latex]\left(-\infty ,7\right][/latex].

Find the domain of the function [latex]f\left(x\right)=\sqrt{5+2x}[/latex].

Can there be functions in which the domain and range do not intersect at all?

Yes. For example, the function [latex]f\left(x\right)=-\frac{1}{\sqrt{x}}[/latex] has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function’s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.

When you are defining the domain of a function, it can help to graph it, especially when you have a rational or a function with an even root.

First, determine the domain restrictions for the following functions, then graph each one to check whether your domain agrees with the graph.

• [latex]f(x) = \sqrt{2x-4}+5[/latex]
• [latex]g(x) = \dfrac{2x+4}{x-1}[/latex]

Next, use an online graphing tool to evaluate your function at the domain restriction you found. What function value does a graphing calculator give you?

How To: Given the formula for a function, determine the domain and range.

• Exclude from the domain any input values that result in division by zero.
• Exclude from the domain any input values that have nonreal (or undefined) number outputs.
• Use the valid input values to determine the range of the output values.
• Look at the function graph and table values to confirm the actual function behavior.

Example: Finding the Domain and Range Using Toolkit Functions

Find the domain and range of [latex]f\left(x\right)=2{x}^{3}-x[/latex].

There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

The domain is [latex]\left(-\infty ,\infty \right)[/latex] and the range is also [latex]\left(-\infty ,\infty \right)[/latex].

Example: Finding the Domain and Range

Find the domain and range of [latex]f\left(x\right)=\dfrac{2}{x+1}[/latex].

We cannot evaluate the function at [latex]-1[/latex] because division by zero is undefined. The domain is [latex]\left(-\infty ,-1\right)\cup \left(-1,\infty \right)[/latex]. Because the function is never zero, we exclude 0 from the range. The range is [latex]\left(-\infty ,0\right)\cup \left(0,\infty \right)[/latex].

Find the domain and range of [latex]f\left(x\right)=2\sqrt{x+4}[/latex].

We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.

[latex]x+4\ge 0\text{ when }x\ge -4[/latex]

The domain of [latex]f\left(x\right)[/latex] is [latex]\left[-4,\infty \right)[/latex].

We then find the range. We know that [latex]f\left(-4\right)=0[/latex], and the function value increases as [latex]x[/latex] increases without any upper limit. We conclude that the range of [latex]f[/latex] is [latex]\left[0,\infty \right)[/latex].

Analysis of the Solution

The graph below represents the function [latex]f[/latex].

Find the domain and range of [latex]f\left(x\right)=-\sqrt{2-x}[/latex].

Domain: [latex]\left(-\infty ,2\right][/latex]   Range: [latex]\left(-\infty ,0\right][/latex]

Determine Domain and Range from a Graph

Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the [latex]x[/latex]-axis. The range is the set of possible output values, which are shown on the [latex]y[/latex]-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values.

We can observe that the graph extends horizontally from [latex]-5[/latex] to the right without bound, so the domain is [latex]\left[-5,\infty \right)[/latex]. The vertical extent of the graph is all range values [latex]5[/latex] and below, so the range is [latex]\left(\mathrm{-\infty },5\right][/latex]. Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.

Example: Finding Domain and Range from a Graph

Find the domain and range of the function [latex]f[/latex].

We can observe that the horizontal extent of the graph is –3 to 1, so the domain of [latex]f[/latex] is [latex]\left(-3,1\right][/latex].

The vertical extent of the graph is 0 to [latex]–4[/latex], so the range is [latex]\left[-4,0\right][/latex].

Example: Finding Domain and Range from a Graph of Oil Production

(credit: modification of work by the U.S. Energy Information Administration )

The input quantity along the horizontal axis is “years,” which we represent with the variable [latex]t[/latex] for time. The output quantity is “thousands of barrels of oil per day,” which we represent with the variable [latex]b[/latex] for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [latex]1973\le t\le 2008[/latex] and the range as approximately [latex]180\le b\le 2010[/latex].

In interval notation, the domain is [latex][1973, 2008][/latex], and the range is about [latex][180, 2010][/latex]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

Given the graph, identify the domain and range using interval notation.

Domain = [latex][1950, 2002][/latex]   Range = [latex][47,000,000, 89,000,000][/latex]

Can a function’s domain and range be the same?

Yes. For example, the domain and range of the cube root function are both the set of all real numbers.

Domain and Range of Toolkit Functions

We will now return to our set of toolkit functions to determine the domain and range of each.

For the constant function [latex]f\left(x\right)=c[/latex], the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant [latex]c[/latex], so the range is the set [latex]\left\{c\right\}[/latex] that contains this single element. In interval notation, this is written as [latex]\left[c,c\right][/latex], the interval that both begins and ends with [latex]c[/latex].

For the identity function [latex]f\left(x\right)=x[/latex], there is no restriction on [latex]x[/latex]. Both the domain and range are the set of all real numbers.

For the absolute value function [latex]f\left(x\right)=|x|[/latex], there is no restriction on [latex]x[/latex]. However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.

For the quadratic function [latex]f\left(x\right)={x}^{2}[/latex], the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.

For the cubic function [latex]f\left(x\right)={x}^{3}[/latex], the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.

For the reciprocal function [latex]f\left(x\right)=\frac{1}{x}[/latex], we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write [latex]\left\{x|\text{ }x\ne 0\right\}[/latex], the set of all real numbers that are not zero.

For the reciprocal squared function [latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex], we cannot divide by [latex]0[/latex], so we must exclude [latex]0[/latex] from the domain. There is also no [latex]x[/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.

For the square root function [latex]f\left(x\right)=\sqrt[]{x}[/latex], we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number [latex]x[/latex] is defined to be positive, even though the square of the negative number [latex]-\sqrt{x}[/latex] also gives us [latex]x[/latex].

For the cube root function [latex]f\left(x\right)=\sqrt[3]{x}[/latex], the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).

Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function [latex]f\left(x\right)=|x|[/latex]. With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude , or modulus , of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.

If we input 0, or a positive value, the output is the same as the input.

[latex]f\left(x\right)=x\text{ if }x\ge 0[/latex]

If we input a negative value, the output is the opposite of the input.

[latex]f\left(x\right)=-x\text{ if }x<0[/latex]

Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to [latex]$10,000[/latex] are taxed at [latex]10%[/latex], and any additional income is taxed at [latex]20%[/latex]. The tax on a total income, [latex] S[/latex] , would be [latex] 0.1S[/latex] if [latex]{S}\le$10,000[/latex]  and [latex]1000 + 0.2 (S - $10,000)[/latex] , if [latex] S> $10,000[/latex] .

A General Note: Piecewise Functions

A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:

[latex] f\left(x\right)=\begin{cases}\text{formula 1 if x is in domain 1}\\ \text{formula 2 if x is in domain 2}\\ \text{formula 3 if x is in domain 3}\end{cases} [/latex]

In piecewise notation, the absolute value function is

[latex]|x|=\begin{cases}\begin{align}x&\text{ if }x\ge 0\\ -x&\text{ if }x<0\end{align}\end{cases}[/latex]

How To: Given a piecewise function, write the formula and identify the domain for each interval.

• Identify the intervals for which different rules apply.
• Determine formulas that describe how to calculate an output from an input in each interval.
• Use braces and if-statements to write the function.

Example: Writing a Piecewise Function

A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, [latex]n[/latex], to the cost, [latex]C[/latex].

Two different formulas will be needed. For [latex]n[/latex]-values under 10, [latex]C=5n[/latex]. For values of [latex]n[/latex] that are 10 or greater, [latex]C=50[/latex].

[latex]C(n)=\begin{cases}\begin{align}{5n}&\hspace{2mm}\text{if}\hspace{2mm}{0}<{n}<{10}\\ 50&\hspace{2mm}\text{if}\hspace{2mm}{n}\ge 10\end{align}\end{cases}[/latex]

The graph is a diagonal line from [latex]n=0[/latex] to [latex]n=10[/latex] and a constant after that. In this example, the two formulas agree at the meeting point where [latex]n=10[/latex], but not all piecewise functions have this property.

Example: Working with a Piecewise Function

A cell phone company uses the function below to determine the cost, [latex]C[/latex], in dollars for [latex]g[/latex] gigabytes of data transfer.

[latex]C\left(g\right)=\begin{cases}\begin{align}{25} \hspace{2mm}&\text{ if }\hspace{2mm}{ 0 }<{ g }<{ 2 }\\ { 25+10 }\left(g - 2\right) \hspace{2mm}&\text{ if }\hspace{2mm}{ g}\ge{ 2 }\end{align}\end{cases}[/latex]

Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.

To find the cost of using 1.5 gigabytes of data, [latex]C(1.5)[/latex], we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.

[latex]C(1.5) = $25[/latex]

To find the cost of using 4 gigabytes of data, [latex]C(4)[/latex], we see that our input of 4 is greater than 2, so we use the second formula.

[latex]C(4)=25 + 10( 4-2) =$45[/latex]

We can see where the function changes from a constant to a shifted and stretched identity at [latex]g=2[/latex]. We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.

How To: Given a piecewise function, sketch a graph.

• Indicate on the [latex]x[/latex]-axis the boundaries defined by the intervals on each piece of the domain.
• For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece. Do not graph two functions over one interval because it would violate the criteria of a function.

Example: Graphing a Piecewise Function

Sketch a graph of the function.

[latex]f\left(x\right)=\begin{cases}\begin{align}{ x }^{2} \hspace{2mm}&\text{ if }\hspace{2mm}{ x }\le{ 1 }\\ { 3 } \hspace{2mm}&\text{ if }\hspace{2mm} { 1 }<{ x }\le 2\\ { x } \hspace{2mm}&\text{ if }\hspace{2mm}{ x }>{ 2 }\end{align}\end{cases}[/latex]

Below are the three components of the piecewise function graphed on separate coordinate systems.

(a) [latex]f\left(x\right)={x}^{2}\text{ if }x\le 1[/latex]; (b) [latex]f\left(x\right)=3\text{ if 1< }x\le 2[/latex]; (c) [latex]f\left(x\right)=x\text{ if }x>2[/latex]

Now that we have sketched each piece individually, we combine them in the same coordinate plane.

Note that the graph does pass the vertical line test even at [latex]x=1[/latex] and [latex]x=2[/latex] because the points [latex]\left(1,3\right)[/latex] and [latex]\left(2,2\right)[/latex] are not part of the graph of the function, though [latex]\left(1,1\right)[/latex] and [latex]\left(2,3\right)[/latex] are.

Graph the following piecewise function.

[latex]f\left(x\right)=\begin{cases}\begin{align}{ x}^{3} \hspace{2mm}&\text{ if }\hspace{2mm}{ x }<{-1 }\\ { -2 } \hspace{2mm}&\text{ if } \hspace{2mm}{ -1 }<{ x }<{ 4 }\\ \sqrt{x} \hspace{2mm}&\text{ if }\hspace{2mm}{ x }>{ 4 }\end{align}\end{cases}[/latex]

You can use an online graphing tool to graph piecewise defined functions. Watch this tutorial video to learn how.

Graph the following piecewise function with an online graphing tool.

Can more than one formula from a piecewise function be applied to a value in the domain?

No. Each value corresponds to one equation in a piecewise formula.

Key Concepts

• The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.
• The domain of a function can be determined by listing the input values of a set of ordered pairs.
• The domain of a function can also be determined by identifying the input values of a function written as an equation.
• Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation.
• For many functions, the domain and range can be determined from a graph.
• An understanding of toolkit functions can be used to find the domain and range of related functions.
• A piecewise function is described by more than one formula.
• A piecewise function can be graphed using each algebraic formula on its assigned subdomain.
• Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
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Domain and range

The domain and range of a function is all the possible values of the independent variable, x , for which y is defined. The range of a function is all the possible values of the dependent variable y . In other words, the domain is the set of values that we can plug into a function that will result in a real y-value; the range is the set of values that the function takes on as a result of plugging in an x value within the domain of the function.

In mathematical terms, given a function f(x), the values that f(x) can take on constitute the range of the function, while all the possible x values constitute the domain. Consider the function f(x) = x 2 .

There are no x-values that will result in the function being undefined and matter what real x-value we plug in, the result will always be a real y-value. Thus, the domain of f(x) = x 2 is all x-values. Then, from looking at the graph or testing a few x-values, we can see that any x-value we plug in will result in a positive y-value. Thus, the range of f(x) = x 2 is all positive y-values.

Notice in the examples above that we described the domain and range using words. While this is possible for all functions, different notations have been developed for expressing domains and ranges in a more concise way. This makes it far easier to express the domains and ranges of multiple functions at a time, particularly as functions get more complicated. Two of these notations are interval notation and set notation.

Interval notation

When using interval notation, domain and range are written as intervals of values. The table below shows the basic symbols used in interval notation and what they mean:

When indicating the domain in interval notation, we need to keep the following in mind:

• The smallest term in the interval is written first, followed by a comma, and then the largest term.
• The first term is the left endpoint and the second term is the right endpoint.
• The endpoints are written between either parentheses or brackets, depending on whether the endpoint is included or not.

Let's look at the same example as above, f(x) = x 2 to see how interval notation is used. Recall that the domain of f(x) = x 2 is all real numbers. In other words, any value from negative infinity to positive infinity will yield a real result. Thus, we can write the domain as:

(-∞, ∞)

We used parentheses rather than brackets around each endpoint because the endpoints are negative and positive infinity, which by definition have no bound. Recall that the range of f(x) = x 2 is all positive y-values, including 0. The range can therefore be written in interval notation as:

[0, ∞)

The union symbol is used when we have a function whose domain or range cannot be described with just a single interval. The union symbol can be read as "or" and it is used throughout various fields of mathematics. In the context of interval notation, it simply means to combine two given intervals. For example, consider the function:

This is the same as our function above, except that it is not defined over the interval (0, 1). The domain of the function is therefore all x-values except those in the interval (0, 1), which we can indicate in interval notation using the union symbol as follows:

(-∞, 0] ∪ [1, ∞)

Note that it is also possible to use multiple union symbols to combine more intervals in the same manner.

Set notation

When using set notation, also referred to as set builder notation, we use inequality symbols to describe the domain and range as a set of values. Like interval notation, there are a number of symbols used in set notation, the most common of which are shown in the table below:

Standard inequality symbols such as , ≥, and so on are also used in set notation.

Using the same example as above, the domain of f(x) = x 2 in set notation is:

{x | x∈ℝ}

The above can be read as "the set of all x such that x is an element of the set of all real numbers." In other words, the domain is all real numbers. We could also write the domain as {x | -∞ < x < ∞}.

The range of f(x) = x 2 in set notation is:

{y | y ≥ 0}

which can be read as "the set of all y such that y is greater than or equal to zero."

Like interval notation, we can also use unions in set builder notation. However, in set notation, rather than using the symbol "∪," we use the word "or" by convention. For example example, given the function

we can write the domain of the above function in set notation as:

{x | x ≤ 0 or x ≥ 1}

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2.3.1: Domain and Range (Exercises)

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For the following exercises, find the domain of each function, expressing answers using interval notation.

20. \(f(x)=\frac{2}{3 x+2}\) 21. \(f(x)=\frac{x-3}{x^{2}-4 x-12}\) 22. \(f(x)=\frac{\sqrt{x-6}}{\sqrt{x-4}}\) 23. Graph this piecewise function: \(f(x)=\left\{\begin{array}{ll}x+1 & x<-2 \\ -2 x-3 & x \geq-2\end{array}\right.\)

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Domain and Range Worksheets

This compilation of domain and range worksheet pdfs provides 8th grade and high school students with ample practice in determining the domain or the set of possible input values (x) and range, the resultant or output values (y) using a variety of exercises with ordered pairs presented on graphs and in table format. Find the domain and range of relations from mapping diagrams, from finite and infinite graphs and more. Get started with our free worksheets.

Write the Domain and Range | Relation - Ordered Pairs

State the domain and range of each relation represented as a set of ordered pairs in Part A and ordered pairs on a graph in Part B of these printable worksheets.

• Download the set

Write the Domain and Range | Relation - Mapping

Determine the domain and range in each of the relations presented in these relation mapping worksheets for grade 8 and high school students. Observe each relation and write the domain (x) and range (y) values in set notation.

Write the Domain and Range | Relation - Table

This batch presents the ordered pairs in tables with input and output columns. Identify the domain and range and write them in ascending order for each of the tables featured in these domain and range pdf worksheets.

Write the Domain and Range | Finite Graph

Observe the extent of the graph horizontally for the domain and the vertical extent of the graph for range and write the smallest and largest values of both in this set of identifying the domain and range from finite graphs worksheets. Use apt brackets to show if the interval is open or closed.

Write the Domain and Range | Infinite Graph

Bolster skills in identifying the domain and range of functions with infinite graphs. Analyze each graph, write the minimum and maximum points for both domain and range. If there is no endpoint, then it can be concluded it is infinite.

Write the Range | Function Rule - Level 1

In this set of pdf worksheets, the function rule is expressed as a linear function and the domain is also provided in each problem. Plug in the values of x in the function rule to determine the range.

Write the Range | Function Rule - Level 2

Substitute the input values or values of the domain in the given quadratic, polynomial, reciprocal or square root functions and determine the output values or range in this section of Level 2 exercises.

Write the Domain and Range | Function - Mixed Review

Test skills acquired with this printable domain and range revision worksheets that provide a mix of absolute, square root, quadratic and reciprocal functions f(x). Determine the domain (x) and plug in the possible x-values to find the range (y).

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1.2 Domain and Range

Learning objectives.

In this section, you will:

• Find the domain of a function defined by an equation.
• Graph piecewise-defined functions.

Horror and thriller movies are both popular and, very often, extremely profitable. When big-budget actors, shooting locations, and special effects are included, however, studios count on even more viewership to be successful. Consider five major thriller/horror entries from the early 2000s— I am Legend , Hannibal , The Ring , The Grudge , and The Conjuring . Figure 1 shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section, we will investigate methods for determining the domain and range of functions such as these.

Finding the Domain of a Function Defined by an Equation

In Functions and Function Notation , we were introduced to the concepts of domain and range . In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.

We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See Figure 2 .

We can write the domain and range in interval notation , which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, they would need to express the interval that is more than 0 and less than or equal to 100 and write ( 0 , 100 ] . ( 0 , 100 ] . We will discuss interval notation in greater detail later.

Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an odd root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.

Before we begin, let us review the conventions of interval notation:

• The smallest term from the interval is written first.
• The largest term in the interval is written second, following a comma.
• Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.
• Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.

See Figure 3 for a summary of interval notation.

Finding the Domain of a Function as a Set of Ordered Pairs

Find the domain of the following function: { ( 2 , 10 ) , ( 3 , 10 ) , ( 4 , 20 ) , ( 5 , 30 ) , ( 6 , 40 ) } { ( 2 , 10 ) , ( 3 , 10 ) , ( 4 , 20 ) , ( 5 , 30 ) , ( 6 , 40 ) } .

First identify the input values. The input value is the first coordinate in an ordered pair . There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.

Find the domain of the function:

{ ( −5 , 4 ) , ( 0 , 0 ) , ( 5 , −4 ) , ( 10 , −8 ) , ( 15 , −12 ) } { ( −5 , 4 ) , ( 0 , 0 ) , ( 5 , −4 ) , ( 10 , −8 ) , ( 15 , −12 ) }

Given a function written in equation form, find the domain.

• Identify the input values.
• Identify any restrictions on the input and exclude those values from the domain.
• Write the domain in interval form, if possible.

Finding the Domain of a Function

Find the domain of the function f ( x ) = x 2 − 1. f ( x ) = x 2 − 1.

The input value, shown by the variable x x in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.

In interval form, the domain of f f is ( − ∞ , ∞ ) . ( − ∞ , ∞ ) .

Find the domain of the function: f ( x ) = 5 − x + x 3 . f ( x ) = 5 − x + x 3 .

Given a function written in an equation form that includes a fraction, find the domain.

• Identify any restrictions on the input. If there is a denominator in the function’s formula, set the denominator equal to zero and solve for x x . If the function’s formula contains an even root, set the radicand greater than or equal to 0, and then solve.
• Write the domain in interval form, making sure to exclude any restricted values from the domain.

Finding the Domain of a Function Involving a Denominator

Find the domain of the function f ( x ) = x + 1 2 − x . f ( x ) = x + 1 2 − x .

When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for x . x .

Now, we will exclude 2 from the domain. The answers are all real numbers where x < 2 x < 2 or x > 2. x > 2. We can use a symbol known as the union, ∪ , ∪ , to combine the two sets. In interval notation, we write the solution: ( −∞ , 2 ) ∪ ( 2 , ∞ ) . ( −∞ , 2 ) ∪ ( 2 , ∞ ) .

In interval form, the domain of f f is ( − ∞ , 2 ) ∪ ( 2 , ∞ ) . ( − ∞ , 2 ) ∪ ( 2 , ∞ ) .

Find the domain of the function: f ( x ) = 1 + 4 x 2 x − 1 . f ( x ) = 1 + 4 x 2 x − 1 .

Given a function written in equation form including an even root, find the domain.

• Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for x . x .
• The solution(s) are the domain of the function. If possible, write the answer in interval form.

Finding the Domain of a Function with an Even Root

Find the domain of the function f ( x ) = 7 − x . f ( x ) = 7 − x .

When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.

Set the radicand greater than or equal to zero and solve for x . x .

Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to 7 , 7 , or ( − ∞ , 7 ] . ( − ∞ , 7 ] .

Find the domain of the function f ( x ) = 5 + 2 x . f ( x ) = 5 + 2 x .

Can there be functions in which the domain and range do not intersect at all?

Yes. For example, the function f ( x ) = − 1 x f ( x ) = − 1 x has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function’s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.

Using Notations to Specify Domain and Range

In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation . For example, { x | 10 ≤ x < 30 } { x | 10 ≤ x < 30 } describes the behavior of x x in set-builder notation. The braces { } { } are read as “the set of,” and the vertical bar | is read as “such that,” so we would read { x | 10 ≤ x < 30 } { x | 10 ≤ x < 30 } as “the set of x -values such that 10 is less than or equal to x , x , and x x is less than 30.”

Figure 5 compares inequality notation, set-builder notation, and interval notation.

To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol, ∪ , ∪ , to combine two unconnected intervals. For example, the union of the sets { 2 , 3 , 5 } { 2 , 3 , 5 } and { 4 , 6 } { 4 , 6 } is the set { 2 , 3 , 4 , 5 , 6 } . { 2 , 3 , 4 , 5 , 6 } . It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

Set-Builder Notation and Interval Notation

Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form { x | statement about  x } { x | statement about  x } which is read as, “the set of all x x such that the statement about x x is true.” For example,

Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,

Given a line graph, describe the set of values using interval notation.

• Identify the intervals to be included in the set by determining where the heavy line overlays the real line.
• At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).
• At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).
• Use the union symbol ∪ ∪ to combine all intervals into one set.

Describing Sets on the Real-Number Line

Describe the intervals of values shown in Figure 6 using inequality notation, set-builder notation, and interval notation.

To describe the values, x , x , included in the intervals shown, we would say, “ x x is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.”

Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.

Given Figure 7 , specify the graphed set in

• ⓑ set-builder notation
• ⓒ interval notation

Finding Domain and Range from Graphs

Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x -axis. The range is the set of possible output values, which are shown on the y -axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See Figure 8 .

We can observe that the graph extends horizontally from −5 −5 to the right without bound, so the domain is [ −5 , ∞ ) . [ −5 , ∞ ) . The vertical extent of the graph is all range values 5 5 and below, so the range is ( −∞ , 5 ] . ( −∞ , 5 ] . Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.

Finding Domain and Range from a Graph

Find the domain and range of the function f f whose graph is shown in Figure 9 .

We can observe that the horizontal extent of the graph is –3 to 1, so the domain of f f is ( − 3 , 1 ] . ( − 3 , 1 ] .

The vertical extent of the graph is 0 to –4, so the range is [ − 4 , 0 ] . [ − 4 , 0 ] . See Figure 10 .

Finding Domain and Range from a Graph of Oil Production

Find the domain and range of the function f f whose graph is shown in Figure 11 .

The input quantity along the horizontal axis is “years,” which we represent with the variable t t for time. The output quantity is “thousands of barrels of oil per day,” which we represent with the variable b b for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as 1973 ≤ t ≤ 2008 1973 ≤ t ≤ 2008 and the range as approximately 180 ≤ b ≤ 2010. 180 ≤ b ≤ 2010.

In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

Given Figure 12 , identify the domain and range using interval notation.

Can a function’s domain and range be the same?

Yes. For example, the domain and range of the cube root function are both the set of all real numbers.

Finding Domains and Ranges of the Toolkit Functions

We will now return to our set of toolkit functions to determine the domain and range of each.

Given the formula for a function, determine the domain and range.

• Exclude from the domain any input values that result in division by zero.
• Exclude from the domain any input values that have nonreal (or undefined) number outputs.
• Use the valid input values to determine the range of the output values.
• Look at the function graph and table values to confirm the actual function behavior.

Finding the Domain and Range Using Toolkit Functions

Find the domain and range of f ( x ) = 2 x 3 − x . f ( x ) = 2 x 3 − x .

There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

The domain is ( − ∞ , ∞ ) ( − ∞ , ∞ ) and the range is also ( − ∞ , ∞ ) . ( − ∞ , ∞ ) .

Finding the Domain and Range

Find the domain and range of f ( x ) = 2 x + 1 . f ( x ) = 2 x + 1 .

We cannot evaluate the function at −1 −1 because division by zero is undefined. The domain is ( − ∞ , −1 ) ∪ ( −1 , ∞ ) . ( − ∞ , −1 ) ∪ ( −1 , ∞ ) . Because the function is never zero, we exclude 0 from the range. The range is ( − ∞ , 0 ) ∪ ( 0 , ∞ ) . ( − ∞ , 0 ) ∪ ( 0 , ∞ ) .

Find the domain and range of f ( x ) = 2 x + 4 . f ( x ) = 2 x + 4 .

We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.

The domain of f ( x ) f ( x ) is [ − 4 , ∞ ) . [ − 4 , ∞ ) .

We then find the range. We know that f ( − 4 ) = 0 , f ( − 4 ) = 0 , and the function value increases as x x increases without any upper limit. We conclude that the range of f f is [ 0 , ∞ ) . [ 0 , ∞ ) .

Figure 22 represents the function f . f .

Find the domain and range of f ( x ) = − 2 − x . f ( x ) = − 2 − x .

Graphing Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function f ( x ) = | x | . f ( x ) = | x | . With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude , or modulus , of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.

If we input 0, or a positive value, the output is the same as the input.

If we input a negative value, the output is the opposite of the input.

Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income S S would be 0.1 S 0.1 S if S ≤ $ 10 , 000 S ≤ $ 10 , 000 and $ 1000 + 0.2 ( S − $ 10 , 000 ) $ 1000 + 0.2 ( S − $ 10 , 000 ) if S > $ 10 , 000. S > $ 10 , 000.

Piecewise Function

A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:

In piecewise notation, the absolute value function is

Given a piecewise function, write the formula and identify the domain for each interval.

• Identify the intervals for which different rules apply.
• Determine formulas that describe how to calculate an output from an input in each interval.
• Use braces and if-statements to write the function.

Writing a Piecewise Function

A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, n , n , to the cost, C . C .

Two different formulas will be needed. For n -values under 10, C = 5 n . C = 5 n . For values of n n that are 10 or greater, C = 50. C = 50.

The function is represented in Figure 23 . The graph is a diagonal line from n = 0 n = 0 to n = 10 n = 10 and a constant after that. In this example, the two formulas agree at the meeting point where n = 10 , n = 10 , but not all piecewise functions have this property.

Working with a Piecewise Function

A cell phone company uses the function below to determine the cost, C , C , in dollars for g g gigabytes of data transfer.

Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.

To find the cost of using 1.5 gigabytes of data, C ( 1.5 ) , C ( 1.5 ) , we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.

To find the cost of using 4 gigabytes of data, C ( 4 ) , C ( 4 ) , we see that our input of 4 is greater than 2, so we use the second formula.

The function is represented in Figure 24 . We can see where the function changes from a constant to a shifted and stretched identity at g = 2. g = 2. We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.

Given a piecewise function, sketch a graph.

• Indicate on the x -axis the boundaries defined by the intervals on each piece of the domain.
• For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece. Do not graph two functions over one interval because it would violate the criteria of a function.

Graphing a Piecewise Function

Sketch a graph of the function.

Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality.

Figure 25 shows the three components of the piecewise function graphed on separate coordinate systems.

Now that we have sketched each piece individually, we combine them in the same coordinate plane. See Figure 26 .

Note that the graph does pass the vertical line test even at x = 1 x = 1 and x = 2 x = 2 because the points ( 1 , 3 ) ( 1 , 3 ) and ( 2 , 2 ) ( 2 , 2 ) are not part of the graph of the function, though ( 1 , 1 ) ( 1 , 1 ) and ( 2 , 3 ) ( 2 , 3 ) are.

Graph the following piecewise function.

Can more than one formula from a piecewise function be applied to a value in the domain?

No. Each value corresponds to one equation in a piecewise formula.

Access these online resources for additional instruction and practice with domain and range.

• Domain and Range of Square Root Functions
• Determining Domain and Range
• Find Domain and Range Given the Graph
• Find Domain and Range Given a Table
• Find Domain and Range Given Points on a Coordinate Plane

1.2 Section Exercises

Why does the domain differ for different functions?

How do we determine the domain of a function defined by an equation?

Explain why the domain of f ( x ) = x 3 f ( x ) = x 3 is different from the domain of f ( x ) = x . f ( x ) = x .

When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?

How do you graph a piecewise function?

For the following exercises, find the domain of each function using interval notation.

f ( x ) = − 2 x ( x − 1 ) ( x − 2 ) f ( x ) = − 2 x ( x − 1 ) ( x − 2 )

f ( x ) = 5 − 2 x 2 f ( x ) = 5 − 2 x 2

f ( x ) = 3 x − 2 f ( x ) = 3 x − 2

f ( x ) = 3 − 6 − 2 x f ( x ) = 3 − 6 − 2 x

f ( x ) = 4 − 3 x f ( x ) = 4 − 3 x

f ( x ) = x 2 + 4 f ( x ) = x 2 + 4

f ( x ) = 1 − 2 x 3 f ( x ) = 1 − 2 x 3

f ( x ) = x − 1 3 f ( x ) = x − 1 3

f ( x ) = 9 x − 6 f ( x ) = 9 x − 6

f ( x ) = 3 x + 1 4 x + 2 f ( x ) = 3 x + 1 4 x + 2

f ( x ) = x + 4 x − 4 f ( x ) = x + 4 x − 4

f ( x ) = x − 3 x 2 + 9 x − 22 f ( x ) = x − 3 x 2 + 9 x − 22

f ( x ) = 1 x 2 − x − 6 f ( x ) = 1 x 2 − x − 6

f ( x ) = 2 x 3 − 250 x 2 − 2 x − 15 f ( x ) = 2 x 3 − 250 x 2 − 2 x − 15

f ( x ) = 5 x − 3 f ( x ) = 5 x − 3

f ( x ) = 2 x + 1 5 − x f ( x ) = 2 x + 1 5 − x

f ( x ) = x − 4 x − 6 f ( x ) = x − 4 x − 6

f ( x ) = x − 6 x − 4 f ( x ) = x − 6 x − 4

f ( x ) = x x f ( x ) = x x

f ( x ) = x 2 − 9 x x 2 − 81 f ( x ) = x 2 − 9 x x 2 − 81

Find the domain of the function f ( x ) = 2 x 3 − 50 x f ( x ) = 2 x 3 − 50 x by:

• ⓐ using algebra.
• ⓑ graphing the function in the radicand and determining intervals on the x -axis for which the radicand is nonnegative.

For the following exercises, write the domain and range of each function using interval notation.

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

f ( x ) = { x + 1 if x < − 2 − 2 x − 3 if x ≥ − 2 f ( x ) = { x + 1 if x < − 2 − 2 x − 3 if x ≥ − 2

f ( x ) = { 2 x − 1 if x < 1 1 + x if x ≥ 1 f ( x ) = { 2 x − 1 if x < 1 1 + x if x ≥ 1

f ( x ) = { x + 1 if x < 0 x − 1 if x > 0 f ( x ) = { x + 1 if x < 0 x − 1 if x > 0

f ( x ) = { 3 if x < 0 x if x ≥ 0 f ( x ) = { 3 if x < 0 x if x ≥ 0

f ( x ) = { x 2       if  x < 0 1 − x   if  x > 0 f ( x ) = { x 2       if  x < 0 1 − x   if  x > 0

f ( x ) = { x 2 x + 2 if x < 0 if x ≥ 0 f ( x ) = { x 2 x + 2 if x < 0 if x ≥ 0

f ( x ) = { x + 1 if x < 1 x 3 if x ≥ 1 f ( x ) = { x + 1 if x < 1 x 3 if x ≥ 1

f ( x ) = { | x | 1 if x < 2 if x ≥ 2 f ( x ) = { | x | 1 if x < 2 if x ≥ 2

For the following exercises, given each function f , f , evaluate f ( −3 ) , f ( −2 ) , f ( −1 ) , f ( −3 ) , f ( −2 ) , f ( −1 ) , and f ( 0 ) . f ( 0 ) .

f ( x ) = { 1 if  x ≤ − 3 0 if  x > − 3 f ( x ) = { 1 if  x ≤ − 3 0 if  x > − 3

f ( x ) = { − 2 x 2 + 3 if  x ≤ − 1 5 x − 7 if  x > − 1 f ( x ) = { − 2 x 2 + 3 if  x ≤ − 1 5 x − 7 if  x > − 1

For the following exercises, given each function f , f , evaluate f ( −1 ) , f ( 0 ) , f ( 2 ) , f ( −1 ) , f ( 0 ) , f ( 2 ) , and f ( 4 ) . f ( 4 ) .

f ( x ) = { 7 x + 3 if x < 0 7 x + 6 if x ≥ 0 f ( x ) = { 7 x + 3 if x < 0 7 x + 6 if x ≥ 0

f ( x ) = { x 2 − 2 if x < 2 4 + | x − 5 | if x ≥ 2 f ( x ) = { x 2 − 2 if x < 2 4 + | x − 5 | if x ≥ 2

f ( x ) = { 5 x if x < 0 3 if 0 ≤ x ≤ 3 x 2 if x > 3 f ( x ) = { 5 x if x < 0 3 if 0 ≤ x ≤ 3 x 2 if x > 3

For the following exercises, write the domain for the piecewise function in interval notation.

f ( x ) = { x + 1  if x < − 2 − 2 x − 3 if x ≥ − 2 f ( x ) = { x + 1  if x < − 2 − 2 x − 3 if x ≥ − 2

f ( x ) = { x 2 − 2  if x < 1 − x 2 + 2 if x > 1 f ( x ) = { x 2 − 2  if x < 1 − x 2 + 2 if x > 1

f ( x ) = { 2 x − 3 − 3 x 2 if x < 0 if x ≥ 2 f ( x ) = { 2 x − 3 − 3 x 2 if x < 0 if x ≥ 2

Graph y = 1 x 2 y = 1 x 2 on the viewing window [ −0.5 , −0.1 ] [ −0.5 , −0.1 ] and [ 0.1 , 0.5 ] . [ 0.1 , 0.5 ] . Determine the corresponding range for the viewing window. Show the graphs.

Graph y = 1 x y = 1 x on the viewing window [ −0.5 , −0.1 ] [ −0.5 , −0.1 ] and [ 0.1 , 0.5 ] . [ 0.1 , 0.5 ] . Determine the corresponding range for the viewing window. Show the graphs.

Suppose the range of a function f f is [ −5 , 8 ] . [ −5 , 8 ] . What is the range of | f ( x ) | ? | f ( x ) | ?

Create a function in which the range is all nonnegative real numbers.

Create a function in which the domain is x > 2. x > 2.

Number Line

• domain\:and\:range\:y=\frac{x^2+x+1}{x}
• domain\:and\:range\:f(x)=x^3
• domain\:and\:range\:f(x)=\ln (x-5)
• domain\:and\:range\:f(x)=\frac{1}{x^2}
• domain\:and\:range\:y=\frac{x}{x^2-6x+8}
• domain\:and\:range\:f(x)=\sqrt{x+3}
• domain\:and\:range\:f(x)=\cos(2x+5)
• domain\:and\:range\:f(x)=\sin(3x)

function-domain-and-range-calculator

• Functions A function basically relates an input to an output, there’s an input, a relationship and an output. For every input...

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