evaluating functions essay

Evaluating Functions

To evaluate a function is to:

Replace ( substitute ) any variable with its given number or expression

Like in this example:

Example: evaluate the function f(x) = 2x+4 for x=5

Just replace the variable "x" with "5":

f( 5 ) = 2× 5 + 4 = 14

Answer: f(5) = 14

More Examples

Here is a function:

• f(x) = 1 − x + x 2

f is just a name, x is just a place-holder.

These are all the same function :

• f(q) = 1 − q + q 2
• w(A) = 1 − A + A 2
• pumpkin(θ) = 1 − θ + θ 2

Evaluate For a Given Value:

Let us evaluate that function for x=3 :

f( 3 ) = 1 − 3 + 3 2 = 1 − 3 + 9 = 7

Evaluate For a Given Expression:

Evaluating can also mean replacing with an expression (such as 3m+1 or v 2 ).

Let us evaluate our function for x=1/r :

f( 1/r ) = 1 − ( 1/r ) + ( 1/r ) 2

Or for x=a−4 :

Another Example

You can use your ability to evaluate functions in other way:

Example: h(x) = 3x 2 + ax − 1

• You are told that h(3) = 8 , can you work out what "a" is?

I recommend putting the substituted values inside parentheses () , so you don't make mistakes.

Example: evaluate the function h(x) = x 2 + 2 for x = −3

Replace the variable "x" with "−3":

h(−3) = (−3) 2 + 2 = 9 + 2 = 11

Without the () you could make a mistake:

h(−3) = −3 2 + 2 = −9 + 2 = −7 (WRONG!)

Also be careful of this:

f(x+a) is not the same as f(x) + f(a)

Example: g(x) = x 2

Different Result!

Evaluating Function

Function notation and how to evaluate a function.

The common notation of a function is usually written as,

Don’t think of this too literally, that is, [latex]f[/latex] is being multiplied to [latex]x[/latex]. Instead, consider this as a mathematical expression which is read as

Functions can also be written in different ways using other variables such as

• [latex]g(x)[/latex], [latex]h(x)[/latex], and [latex]k(x)[/latex]

In addition, functions may take other input values other than [latex]x[/latex].

• [latex]f(a)[/latex], [latex]h(r)[/latex], and [latex]k(m)[/latex]

The key idea is always to remember that the variable outside the parenthesis is the “ name ” of the function, while the variable inside the parenthesis is the input value of the function.

For instance, the following is called function [latex]k[/latex] with an input value of [latex]m[/latex].

Basic Examples of Evaluating Functions

Example 1: Evaluate the function.

This is the normal notation of function where the function is [latex]f[/latex] while the input value is [latex]x[/latex]. To evaluate a function, what we want is to substitute every instance of [latex]x[/latex] in the expression and then simplify.

Since [latex]x = – 1[/latex] , we substitute this value in the function and simplify. In doing so, we get a solution that looks like this.

Example 2: Evaluate the function.

Observe that the function here is [latex]h[/latex] and the input value is [latex]k[/latex]. Just like in our previous example, we want to substitute whatever the numerical value assigned to [latex]k[/latex] into the given function, and simplify.

Since [latex]k = 3[/latex], your solution should look similar to this

Example 3: Evaluate each value of [latex]x[/latex] in the table below using the function below. Plot the points in the [latex]xy[/latex]-axis and connect the dots to reveal the graph of the function.

Since there are seven [latex]x[/latex]-inputs, that means we will evaluate the function seven times as well. Try working this out on your own then come back to check your answers.

If you have done it correctly, these are the values:

We can now place those output values in the table.

Think of the output values of the function [latex]f\left( x \right)[/latex] as the [latex]y[/latex]-values. This is how the graph looks like on the [latex]xy[/latex]-axis.

Intermediate Examples of Evaluating Functions

Example 4: Given that [latex]g\left( x \right) = {x^2} – 3x + 1[/latex], find [latex]g\left( {2x – 1} \right)[/latex].

In previous examples, we have been evaluating a function by a number. This time the input value is no longer a fixed numerical value, but instead an expression. It might look complicated but the procedure remains the same.

We will replace every instance of [latex]x[/latex] in [latex]g\left( x \right)[/latex] by the input value which is [latex]2x – 1[/latex]. Simplify by squaring the binomial , applying the distributive property , and combining like terms .

Example 5: Given that [latex]p\left( x \right) = {{4x – 1} \over x}[/latex] , evaluate [latex]p\left( 1 \right) – p\left( { – 1} \right)[/latex].

The problem may look intimidating at first, but once we analyze it and apply what we already know on how to evaluate functions, this shouldn’t be that bad!

What we need to do here is to evaluate the function at [latex]x = 1[/latex] then subtract by the value of the function when evaluated at [latex]x = – \,1[/latex].

Be very careful when you substitute the values and during the simplification process. If you are not careful in every step, it is very easy to commit mistakes when you add, subtract, multiply, or divide positive and negative numbers.

Advanced Example of Applying the Concept of Evaluating Functions

Example 6:   If [latex]f\left( 2 \right) = 9[/latex], find the value of [latex]a[/latex] in the function below.

In the equation, [latex]f\left( 2 \right) = 9[/latex], we are told that if the input of the function is [latex]2[/latex]; the output of the function will be [latex]9[/latex]. Since the function is given to us, our first move is to at least substitute the value of [latex]2[/latex] and then simplify. This is what we’ll get.

The output of the function after evaluating at [latex]x = 2[/latex] is [latex]17 + 2a[/latex]. Remember, we are also told that the output is [latex]9[/latex] using the given equation [latex]f\left( 2 \right) = 9[/latex]. Therefore what we need to do now is set them equal to each other, and solve the linear equation for the unknown value of [latex]a[/latex].

Let’s verify if the value of [latex]a = – \,4[/latex] in [latex]f(x) = 6{x^2} + ax – 7[/latex] can make the given condition [latex]f\left( 2 \right) = 9[/latex] to be a true statement.

It’s true! Hence, we have successfully solved the correct value of [latex]a[/latex].

Evaluate Functions

How do you evaluate functions.

The same way that you substitute values into equations!

What is the value of $$ x $$ given the equation $$ y = 2x $$ when $$ x = 5 $$ ?

Substitute '5' in for x :

The one new aspect of function notation is the emphasis on input and output .

What is the value of $$ x $$ given the equation $$ y = x-5 $$ when $$ x = 7 $$ ?

Again, this new way involves an input and output .

Function Machine

You can think of $$ f(x)$$ as a function machine. The function machine, or $$ f(x)$$ , takes input inside . The machine processes this input and produces an output value

Practice Problems

Let $$ k(x) =3x $$ .

Evaluate $$ k (5) $$ .

Identify all of the occurrences of 'x' and substitute the input in

$$ k(\blue{x }) =3 \blue{x } \\ k(\blue{5 }) =3\cdot \blue{5 } $$

Compute result.

$$ k(\blue5 ) =3\cdot \blue 5 \\ = \red {15} $$

$$ k(\blue {input }) =\red {output} \\ k(\blue 5) =\red {15} $$

Let $$ g(x) =3x^2 + 7x $$ .

Evaluate $$ g(4) $$ .

$$ g(\blue x ) =3 \blue x ^2 + 7 \blue x \\ g(\blue 4 ) =3\cdot \blue 4 ^2 + 7\cdot \blue 4 $$

$$ g(\blue 4 ) =3\cdot \blue 4 ^2 + 7\cdot \blue 4 \\ g(\blue 4 ) = \red{76} $$

$$ g(\blue {input }) =\red {output} \\ g(\blue 4) =\red {76} $$

Let $$ h(x) =\sqrt{x^3 -4}-|x| $$ .

Evaluate $$ h(5) $$ .

$$ h(\blue x ) = \sqrt{\blue x ^3 -4}-| \blue{x }| \\ h(\blue {5 }) = \sqrt{\blue 5 ^3 - 4 } - | \blue 5 | $$

$$ h(\blue 5 ) = \sqrt{\blue 5 ^3 -4}-|\blue 5 | \\ h(\blue 5 ) =50 \\ h(\blue 5) = \red 6 $$

$$ h(\blue {input }) =\red {output} \\ h(\blue 5) =\red 6 $$

Let $$ f(x) = -3x^2 + 5x - 1 $$ .

Evaluate $$ f(6) $$ .

$$ f(\blue x ) = -3 \blue x ^2 + 5\blue x - 1 \\ f(\blue 6 ) = -3 \cdot \blue 6 ^2 + 5\cdot \blue 6 - 1 $$

$$ f(\blue 6 ) = -3 \cdot \blue 6^2 + 5\cdot \blue 6 - 1 \\ f(\blue 6) = \red{-79} $$

$$ f(\blue {input }) =\red {output} \\ f(\blue 6) =\red {-79} $$

The height in meters of a projectile at t seconds can be found by the function $$ h(t) = -5t^2 + 40t + 1.2 $$ .

Find the height of the projectile 4 seconds after it is launched.

$$ h(\blue t ) = -5\blue t^2 + 40 \blue t + 1.2 \\ h(\blue 4 ) = -5 \cdot \blue 4 ^2 + 40 \cdot \blue 4 + 1.2 $$

$$ h(\blue 4 ) = -5 \cdot \blue 4^2 + 40 \cdot \blue 4 + 1.2 \\ h(\blue 4 ) = \red{81.2} $$

$$ h(\blue {input }) =\red {output} \\ h(\blue 4) =\red {81.2} $$

A substance has a half life of 26 years. The amount of remaining substance in grams after t years can be found by the function $$ h(t) = 250 (0.5)^{ \frac{t}{25} } $$ .

How much substance remains after 98 years?

$$ h(\blue t) = 250 (0.5)^{ \frac{\blue t}{25} } \\ h(\blue { 98 }) = 250 (0.5)^{ \frac{\blue{98}}{25} } $$

$$ h(\blue{98}) = 250 (0.5)^{ \frac{\blue{98}}{25} } \\ h(\blue{98}) = \red{16.5159} $$

$$ h(\blue {input }) =\red {output} \\ h(\blue {98}) =\red {16.5159} $$

Here is a picture of graph of projectile's path with the point $$ (\color{blue} {t}, \color{red} {h(t)}) (\color{blue} {98} , \color{red} {16.5159}) $$ : .

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Evaluating a function means finding the value of f(x) =… or y =… that corresponds to a given value of x. To do this, simply replace all the x variables with whatever x has been assigned. For example, if we are asked to evaluate f(4), then x has been assigned the value of 4.

Example: Given that f(x) = 3x + 6, find f(2)

Solution: This means we will evaluate the function when x has been assigned the value of 2. The first step is to replace every x with 2. Then evaluate the function by following order of operations (BEDMAS).

f(x) = 3x + 6

So, we have :

f(2) = 3(2) + 6

        = 6 + 6

        = 12

Example: Given the function f(x) = 5x 2 - 2√ x + 7 , evaluate f(-3).

Solution: Substitute -3 in for x and evaluate.

f(-3) = 5(-3) 2 - 2√ -3 +7

         = 5(9) -2√ 4

         = 45 – 4

         = 41

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Study Guides > MATH 0123

Evaluate functions, learning outcomes.

• Given a function described by an equation, find function values (outputs) for numerical inputs
• Given a function described by an equation, find function values (outputs) for variable inputs

[latex]f(x)=4x+1\\f(2)=4(2)+1=8+1=9[/latex]

[latex]f(5)=3(5)-4[/latex]

[latex]f(5)=15-4\\f(5)=11[/latex]

[latex]p(−3)=2(−3)^{2}+5[/latex]

[latex]p(−3)=2(9)+5\\p(−3)=18+5\\p(−3)=23[/latex]

[latex]f(0)=|4(0)-3|=|-3|=3\\f(0)=3[/latex]

[latex]f(2)=|4(2)-3|=|5|=5\\f(2)=5[/latex]

[latex]f(−1)=|4(-1)-3|=|-7|=7\\f(-1)=7[/latex]

Variable Inputs

Answer: This problem is asking you to evaluate the function for b . This means substitute b in the equation for x. [latex-display]f(b)=3b^{2}+2b+1[/latex-display] (That is all—you are done.)

Answer: This time, you substitute [latex](h+1)[/latex] into the equation for x. [latex]f(h+1)=4(h+1)+1[/latex]   Use the distributive property on the right side, and then combine like terms to simplify. [latex-display]f(h+1)=4h+4+1=4h+5[/latex-display] Given [latex]f(x)=4x+1[/latex], [latex]f(h+1)=4h+5[/latex].

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Module 5: Function Basics

Evaluating and solving functions, learning outcomes.

• Evaluate and solve functions in algebraic form.
• Evaluate functions given tabular or graphical data.

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function [latex]f\left(x\right)=5 - 3{x}^{2}[/latex] can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

How To: EVALUATE A FUNCTION Given ITS FORMula.

• Replace the input variable in the formula with the value provided.
• Calculate the result.

Example: Evaluating Functions

Given the function [latex]h\left(p\right)={p}^{2}+2p[/latex], evaluate [latex]h\left(4\right)[/latex].

To evaluate [latex]h\left(4\right)[/latex], we substitute the value 4 for the input variable [latex]p[/latex] in the given function.

[latex]\begin{align}h\left(p\right)&={p}^{2}+2p \\ h\left(4\right)&={\left(4\right)}^{2}+2\left(4\right) \\ &=16+8 \\ &=24 \end{align}[/latex]

Therefore, for an input of 4, we have an output of 24 or [latex]h(4)=24[/latex].

Watch the video below for more examples of evaluating a function for specific values of the input.

evaluating functions

When evaluating functions, it’s handy to wrap the input variable in parentheses before making the substitution.

Ex. Given [latex]f(x)=x^2 - 8[/latex], find [latex]f(-3)[/latex]

[latex]\begin{align}f(x)&=(x)^2 - 8 \\ &= (-3)^2 - 8 \\ &= 9 - 8 \\ &= 1\end{align}[/latex]

The value of the function [latex]f(x)=x^2 - 8[/latex], at the input [latex]x=-3[/latex], is [latex]1[/latex].

Example: Evaluating Functions at Specific Values

For the function, [latex]f\left(x\right)={x}^{2}+3x - 4[/latex], evaluate each of the following.

• [latex]f\left(2\right)[/latex]
• [latex]f(a)[/latex]
• [latex]f(a+h)[/latex]
• [latex]\dfrac{f\left(a+h\right)-f\left(a\right)}{h}[/latex]

Replace the [latex]x[/latex] in the function with each specified value.

• Because the input value is a number, 2, we can use algebra to simplify. [latex]\begin{align}f\left(2\right)&={2}^{2}+3\left(2\right)-4 \\ &=4+6 - 4 \\ &=6\hfill \end{align}[/latex]
• In this case, the input value is a letter so we cannot simplify the answer any further. [latex]f\left(a\right)={a}^{2}+3a - 4[/latex]
• With an input value of [latex]a+h[/latex], we must use the distributive property. [latex]\begin{align}f\left(a+h\right)&={\left(a+h\right)}^{2}+3\left(a+h\right)-4 \\[2mm] &={a}^{2}+2ah+{h}^{2}+3a+3h - 4 \end{align}[/latex]

and we know that

Now we combine the results and simplify.

[latex]\begin{align}\dfrac{f\left(a+h\right)-f\left(a\right)}{h}&=\dfrac{\left({a}^{2}+2ah+{h}^{2}+3a+3h - 4\right)-\left({a}^{2}+3a - 4\right)}{h} \\[2mm] &=\dfrac{2ah+{h}^{2}+3h}{h}\\[2mm] &=\frac{h\left(2a+h+3\right)}{h}&&\text{Factor out }h. \\[2mm] &=2a+h+3&&\text{Simplify}.\end{align}[/latex]

Given the function [latex]g\left(m\right)=\sqrt{m - 4}[/latex], evaluate [latex]g\left(5\right)[/latex].

[latex]g\left(5\right)=\sqrt{5 - 4}=1[/latex]

In addition to evaluating functions for a particular input, we can also solve functions for the input that creates a particular output.

Example: Solving Functions

Given the function [latex]h\left(p\right)={p}^{2}+2p[/latex], solve for [latex]h\left(p\right)=3[/latex].

[latex]\begin{align}&h\left(p\right)=3\\ &{p}^{2}+2p=3 &&\text{Substitute the original function }h\left(p\right)={p}^{2}+2p. \\ &{p}^{2}+2p - 3=0 &&\text{Subtract 3 from each side}. \\ &\left(p+3\text{)(}p - 1\right)=0 &&\text{Factor}. \end{align}[/latex]

If [latex]\left(p+3\right)\left(p - 1\right)=0[/latex], either [latex]\left(p+3\right)=0[/latex] or [latex]\left(p - 1\right)=0[/latex] (or both of them equal 0). We will set each factor equal to 0 and solve for [latex]p[/latex] in each case.

[latex]\begin{align}&p+3=0, &&p=-3 \\ &p - 1=0, &&p=1\hfill \end{align}[/latex]

This gives us two solutions. The output [latex]h\left(p\right)=3[/latex] when the input is either [latex]p=1[/latex] or [latex]p=-3[/latex].

We can also verify by graphing as in Figure 5. The graph verifies that [latex]h\left(1\right)=h\left(-3\right)=3[/latex] and [latex]h\left(4\right)=24[/latex].

How To: Solve a Function.

• Replace the output in the formula with the value provided.
• Solve for the input variable that makes the statement true.

The next video shows another example of how to solve a function.

Given the function [latex]g\left(m\right)=\sqrt{m - 4}[/latex], solve [latex]g\left(m\right)=2[/latex].

[latex]m=8[/latex]

Evaluating Functions Expressed in Formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation [latex]2n+6p=12[/latex] expresses a functional relationship between [latex]n[/latex] and [latex]p[/latex]. We can rewrite it to decide if [latex]p[/latex] is a function of [latex]n[/latex].

functions, Equations, and formulas

We’ve seen that an equation such as [latex]ax+by=c[/latex] can be written in a different form by solving the equation for one of the variables. If we solve this linear equation for  y  it can be written in the slope-intercept form of a line, [latex]y = mx+b[/latex].

Certain formulas can be written in function form by solving for one of the variables. For instance, can you see how to solve the formula for a rectangle having a perimeter of 21 feet, [latex]21 = 2l + 2w[/latex], for length?

[latex]\begin{align} 21 &= 2l + 2w \\ 21 - 2w &= 2l \\ \dfrac{21-2w}{2} &= l \end{align}[/latex]

We can now declare a function, [latex]l = f(x)[/latex] that returns an output length for a rectangle having a perimeter of 21 feet based on different width inputs.

How To: Given a function in equation form, write its algebraic formula.

• Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves only the input variable.
• Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.

Example: Finding an Equation of a Function

Express the relationship [latex]2n+6p=12[/latex] as a function [latex]p=f\left(n\right)[/latex], if possible.

To express the relationship in this form, we need to be able to write the relationship where [latex]p[/latex] is a function of [latex]n[/latex], which means writing it as [latex]p=[/latex] expression involving [latex]n[/latex].

[latex]\begin{align}&2n+6p=12\\[1mm] &6p=12 - 2n &&\text{Subtract }2n\text{ from both sides}. \\[1mm] &p=\frac{12 - 2n}{6} &&\text{Divide both sides by 6 and simplify}. \\[1mm] &p=\frac{12}{6}-\frac{2n}{6} \\[1mm] &p=2-\frac{1}{3}n \end{align}[/latex]

Therefore, [latex]p[/latex] as a function of [latex]n[/latex] is written as

[latex]p=f\left(n\right)=2-\frac{1}{3}n[/latex]

Analysis of the Solution

It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.

Watch this video to see another example of how to express an equation as a function.

Sometimes a relationship between variables cannot be expressed as a function. See the example below for more information.

Example: Expressing the Equation of a Circle as a Function

Does the equation [latex]{x}^{2}+{y}^{2}=1[/latex] represent a function with [latex]x[/latex] as input and [latex]y[/latex] as output? If so, express the relationship as a function [latex]y=f\left(x\right)[/latex].

First we subtract [latex]{x}^{2}[/latex] from both sides.

[latex]{y}^{2}=1-{x}^{2}[/latex]

We now try to solve for [latex]y[/latex] in this equation.

[latex]\begin{align}y&=\pm \sqrt{1-{x}^{2}} \\[1mm] &=\sqrt{1-{x}^{2}}\hspace{3mm}\text{and}\hspace{3mm}-\sqrt{1-{x}^{2}} \end{align}[/latex]

We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function [latex]y=f\left(x\right)[/latex].

If [latex]x - 8{y}^{3}=0[/latex], express [latex]y[/latex] as a function of [latex]x[/latex].

Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?

Yes, this can happen. For example, given the equation [latex]x=y+{2}^{y}[/latex], if we want to express [latex]y[/latex] as a function of [latex]x[/latex], there is no simple algebraic formula involving only [latex]x[/latex] that equals [latex]y[/latex]. However, each [latex]x[/latex] does determine a unique value for [latex]y[/latex], and there are mathematical procedures by which [latex]y[/latex] can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for [latex]y[/latex] as a function of [latex]x[/latex], even though the formula cannot be written explicitly.

Evaluating a Function Given in Tabular Form

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.

The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See the table below.

At times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P[/latex].

The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function [latex]P[/latex] at the input value of “goldfish.” We would write [latex]P\left(\text{goldfish}\right)=2160[/latex]. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function [latex]P[/latex] seems ideally suited to this function, more so than writing it in paragraph or function form.

How To: Given a function represented by a table, identify specific output and input values.

• Find the given input in the row (or column) of input values.
• Identify the corresponding output value paired with that input value.
• Find the given output values in the row (or column) of output values, noting every time that output value appears.
• Identify the input value(s) corresponding to the given output value.

Example: Evaluating and Solving a Tabular Function

Using the table below,

• Evaluate [latex]g\left(3\right)[/latex].
• Solve [latex]g\left(n\right)=6[/latex].

• Evaluating [latex]g\left(3\right)[/latex] means determining the output value of the function [latex]g[/latex] for the input value of [latex]n=3[/latex]. The table output value corresponding to [latex]n=3[/latex] is 7, so [latex]g\left(3\right)=7[/latex].
• Solving [latex]g\left(n\right)=6[/latex] means identifying the input values, [latex]n[/latex], that produce an output value of 6. The table below shows two solutions: [latex]n=2[/latex] and [latex]n=4[/latex].

When we input 2 into the function [latex]g[/latex], our output is 6. When we input 4 into the function [latex]g[/latex], our output is also 6.

Using the table from the previous example, evaluate [latex]g\left(1\right)[/latex] .

Finding Function Values from a Graph

Ordered pairs of inputs and outputs.

We can view a function as a set of inputs and their corresponding outputs. That is, we can see a function as a set of ordered pairs, [latex]\left(x, y \right).[/latex]

Remember that, in function notation, [latex]y = f(x)[/latex], so the ordered pairs containing inputs and outputs can be written in the form of ( input ,  output ) or [latex]\left(x, f(x)\right)[/latex].

Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

Example: Reading Function Values from a Graph

Given the graph below,

• Evaluate [latex]f\left(2\right)[/latex].
• Solve [latex]f\left(x\right)=4[/latex].

Using the graph, solve [latex]f\left(x\right)=1[/latex].

[latex]x=0[/latex] or [latex]x=2[/latex]

• Graph the function [latex]f(x) = -\frac{1}{2}x^2+x+4[/latex] using function notation.
• Evaluate the function at [latex]x=1[/latex]
• Make a table of values that references the function. Include at least the interval [latex][-5,5][/latex] for [latex]x[/latex]-values.
• Solve the function for [latex]f(0)[/latex]
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Ultimate guide on writing an effective evaluation essay – tips, examples, and guidelines.

Are you puzzled when it comes to writing an evaluation essay? In this guide, we will provide you with all the essential information you need to master the art of crafting a compelling appraisal composition. Whether you are new to this type of writing or just looking to refine your skills, this comprehensive manual will equip you with the necessary tools and techniques to excel. From understanding the purpose and structure of an evaluation essay to exploring various tips and examples, this guide has got you covered.

An evaluation essay is a piece of writing that aims to assess the value or quality of a particular subject or phenomenon. It involves analyzing a topic, presenting your judgment or opinion on it, and providing evidence or examples to support your claims. This type of essay requires critical thinking, research, and effective communication skills to present a well-balanced evaluation.

Throughout this guide, we will delve into the nitty-gritty of writing an evaluation essay. We will start by discussing the key elements that make up a successful evaluation essay, such as establishing clear criteria, conducting thorough research, and adopting a structured approach. Additionally, we will explore practical tips and strategies to help you gather relevant information, organize your thoughts, and present a persuasive argument. To illustrate these concepts, we will provide you with a range of examples covering various topics and subjects.

Tips for Writing a Top-Notch Evaluation Essay

When it comes to crafting a high-quality evaluation essay, there are several key tips to keep in mind. By following these guidelines, you can ensure that your essay stands out and effectively evaluates the subject matter at hand.

1. Be objective and unbiased: A top-notch evaluation essay should approach the topic with an unbiased and objective perspective. Avoid personal bias or overly emotional language, and instead focus on presenting an honest and well-balanced evaluation of the subject.

2. Provide clear criteria: To effectively evaluate something, it’s important to establish clear criteria or standards by which to assess it. Clearly define the criteria you will be using and explain why these specific factors are essential in evaluating the subject. This will help provide structure to your essay and ensure that your evaluation is thorough and comprehensive.

3. Support your evaluation with evidence: In order to make a convincing argument, it’s crucial to support your evaluation with solid evidence. This can include examples, statistics, expert opinions, or any other relevant information that strengthens your claims. By providing strong evidence, you can enhance the credibility of your evaluation and make it more persuasive.

4. Consider multiple perspectives: A well-rounded evaluation takes into account multiple perspectives on the subject matter. Acknowledge and address counterarguments or differing opinions, and provide thoughtful analysis and reasoning for your stance. This demonstrates critical thinking and a comprehensive evaluation of the topic.

5. Use clear and concise language: Clarity is vital in an evaluation essay. Use clear and concise language to express your thoughts and ideas, avoiding unnecessary jargon or complex vocabulary. Your essay should be accessible to a wide audience and easy to understand, allowing your evaluation to be conveyed effectively.

6. Revise and edit: Don’t neglect the importance of revising and editing your essay. Take the time to review your work and ensure that your evaluation is well-structured, coherent, and error-free. Pay attention to grammar, spelling, and punctuation, as these details can greatly impact the overall quality of your essay.

7. Conclude with a strong summary: For a top-notch evaluation essay, it’s important to conclude with a strong and concise summary of your evaluation. Restate your main points and findings, providing a clear and memorable conclusion that leaves a lasting impression on the reader.

By following these tips, you can enhance your writing skills and create a top-notch evaluation essay that effectively assesses and evaluates the subject matter at hand.

Choose a Relevant and Engaging Topic

When it comes to writing an evaluation essay, one of the most important aspects is selecting a topic that is both relevant and engaging. The topic you choose will determine the focus of your essay and greatly impact the overall quality of your writing. It is crucial to choose a topic that not only interests you but also captivates your audience.

When selecting a topic, consider the subject matter that you are knowledgeable or passionate about. This will enable you to provide a well-informed evaluation and maintain your readers’ interest throughout your essay. Additionally, choose a topic that is relevant in today’s society or has a direct impact on your target audience. This will ensure that your evaluation essay has a practical and meaningful purpose.

Furthermore, it is essential to select a topic that is controversial or debatable. This will allow you to present different perspectives and arguments to support your evaluation. By choosing a topic that sparks discussions and debates, you can engage your readers and encourage them to think critically about the subject matter.

In conclusion, choosing a relevant and engaging topic is crucial for writing an effective evaluation essay. By selecting a topic that interests you, appeals to your readers, and is relevant to society, you can ensure that your essay is engaging and impactful. Remember to choose a topic that is controversial or debatable to provide a comprehensive evaluation and encourage critical thinking among your audience.

Develop a Strong Thesis Statement

Crafting an impactful thesis statement is an essential aspect of writing an evaluation essay. The thesis statement serves as the main argument or claim that you will be supporting throughout your essay. It encapsulates the central idea and sets the tone for the rest of the paper.

When developing your thesis statement, it is crucial to be clear, concise, and specific. It should provide a clear indication of your stance on the subject matter being evaluated while also highlighting the main criteria and evidence that will be discussed in the body paragraphs. A strong thesis statement should be thought-provoking and hook the reader’s attention, compelling them to continue reading.

To build a strong thesis statement, you need to engage in a careful analysis of the topic or subject being evaluated. Consider the various aspects that you will be assessing and select the most significant ones to include in your argument. Your thesis statement should be focused and arguable, allowing for a clear position on the matter.

Additionally, it is crucial to avoid vague or general statements in your thesis. Instead, aim for specificity and clarity. By clearly stating your evaluation criteria, you provide a roadmap for the reader to understand what aspects you will be analyzing and what conclusions you intend to make.

Furthermore, a strong thesis statement should be supported by evidence and examples. You should be able to provide concrete support for your evaluation through relevant facts, statistics, or expert opinions. This strengthens the credibility and persuasiveness of your argument, making your thesis statement more compelling.

In summary, developing a strong thesis statement is a critical step in writing an evaluation essay. It sets the foundation for your argument, guiding your analysis and providing a clear direction for the reader. By being clear, concise, specific, and well-supported, your thesis statement helps you create a persuasive and impactful evaluation essay.

Provide Clear and Concise Criteria for Evaluation

One of the most important aspects of writing an evaluation essay is providing clear and concise criteria for evaluation. In order to effectively evaluate a subject or topic, it is essential to establish specific standards or benchmarks that will be used to assess its performance or quality.

When establishing criteria for evaluation, it is crucial to be thorough yet succinct. Clear criteria enable the reader to understand the basis upon which the evaluation is made, while concise criteria ensure that the evaluation remains focused and impactful.

There are several strategies you can employ to provide clear and concise criteria for evaluation. One approach is to define specific attributes or characteristics that are relevant to the subject being evaluated. For example, if you are evaluating a restaurant, you might establish criteria such as the quality of the food, the level of service, and the ambience of the establishment.

Another strategy is to utilize a scoring system or rating scale to assess the subject. This can help provide a more quantitative evaluation by assigning numerical values to different aspects of the subject. For instance, a movie review might use a rating scale of 1 to 5 to evaluate the acting, plot, and cinematography of the film.

In addition to defining specific attributes or using a scoring system, it is important to provide examples or evidence to support your evaluation. This can help make your criteria more concrete and relatable to the reader. For instance, if you are evaluating a car, you could provide examples of its fuel efficiency, handling performance, and safety features.

By providing clear and concise criteria for evaluation, you can effectively communicate your assessment to the reader and support your conclusions. This will help ensure that your evaluation essay is well-structured, informative, and persuasive.

Support Your Evaluation with Solid Evidence

When writing an evaluation essay, it is crucial to support your evaluations with solid evidence. Without proper evidence, your evaluation may appear weak and unsubstantiated. By providing strong evidence, you can convince your readers of the validity of your evaluation and make a compelling argument.

One effective way to support your evaluation is by using concrete examples. These examples can be specific instances or cases that illustrate the strengths or weaknesses of the subject being evaluated. By presenting real-life examples, you can provide tangible evidence and make your evaluation more persuasive.

Another way to support your evaluation is by referring to expert opinions or research studies. These external sources can add credibility to your evaluation and demonstrate that your assessment is based on sound knowledge and expertise. Citing respected experts or referencing reputable studies can enhance the validity of your evaluation and make it more convincing.

In addition to concrete examples and expert opinions, statistical data can also be a powerful tool for supporting your evaluation. Numbers and statistics can provide objective evidence and strengthen your evaluation by adding a quantitative dimension to your argument. By citing relevant statistics, you can add weight to your evaluations and demonstrate the magnitude of the subject’s strengths or weaknesses.

Furthermore, it is important to consider counterarguments and address them in your evaluation. By acknowledging opposing viewpoints and addressing them effectively, you can strengthen your own evaluation and demonstrate a thorough understanding of the subject. This approach shows that you have considered different perspectives and have arrived at a well-rounded evaluation.

In conclusion, supporting your evaluation with solid evidence is essential to writing a persuasive evaluation essay. By using concrete examples, expert opinions, statistical data, and addressing counterarguments, you can bolster the validity and strength of your evaluation. Remember to present your evidence clearly and logically, making your evaluation more compelling and convincing to your readers.

Use a Structured Format to Organize Your Essay

When writing an evaluation essay, it is important to use a structured format to organize your thoughts and arguments. This will help you present your ideas in a clear and logical manner, making it easier for your reader to follow along and understand your points. By using a structured format, you can ensure that your essay flows smoothly and effectively communicates your evaluation.

One effective way to structure your evaluation essay is to use a table format. This allows you to present your evaluation criteria and supporting evidence in a concise and organized manner. By using a table, you can easily compare and contrast different aspects of the subject being evaluated, making it easier for your reader to grasp the overall evaluation.

In addition to using a table format, you should also follow a logical structure within each section of your essay. Start with a clear introduction, where you introduce the subject you are evaluating and provide some background information. Then, present your evaluation criteria and explain why these criteria are important for assessing the subject. Next, provide specific examples and evidence to support your evaluation, using the table format as a guide. Finally, end your essay with a strong conclusion that summarizes your evaluation and reinforces your main points.

By using a structured format, you can effectively organize your evaluation essay and present your ideas in a clear and concise manner. This will make your essay more engaging and persuasive, and help your reader understand and appreciate your evaluation.

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Course: Algebra 1   >   Unit 8

• What is a function?
• Worked example: Evaluating functions from equation
• Evaluate functions
• Worked example: Evaluating functions from graph
• Evaluating discrete functions
• Evaluate functions from their graph
• Worked example: evaluating expressions with function notation

Evaluate function expressions

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍