Evaluate, combine, and compose functions

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Difficulty: Beginner
Estimated time: 35 minutes
Domains: Algebra Advanced Math
Techniques: EvaluationFunctions

What you’ll learn

Identify the input and output in function notation.
Define a function once and evaluate it at different inputs.
Calculate expressions that combine function outputs.
Define and evaluate a new function built from other functions.
Compose functions from the inside out.
Interpret a function value with the correct contextual unit.
Choose a quick hand substitution when Desmos would add work.

Prerequisites

Introduction to Desmos: your SAT calculator0%
Enter parentheses, powers, fractions, and expressions accurately.

Why this matters on the SAT

Turn function notation into a direct calculation

SAT function questions often ask for an output hidden inside notation such as $f(4)$ or $2f(3)-g(3)$ . Some harder questions place one function inside another. Desmos understands each form directly. Define every function once, then enter exactly what the question asks for.

SAT example

The functions $f$ and $g$ are defined by

f(x)=x^2+3

and

g(x)=4x-1.

What is the value of $2f(-2)-g(-2)$ ?

A
$-23$
B
$5$
C
$14$
D
$23$

Fast Desmos solution

Enter f(x)=x^2+3 and g(x)=4x-1. Then enter 2f(-2)-g(-2). Desmos returns $23$ , so the answer is D.

The notation shows the calculation order. Desmos handles the substitution, but you still need to identify the requested expression and interpret the result.

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Desmos evaluates both functions and combines their outputs in one expression.

When should you evaluate a function in Desmos?

Desmos is strongest when function notation turns one substitution into a longer, nested, or repeated calculation.

Begin with Desmos when…

a function rule is given and the question asks for one or more outputs;
the input is negative, fractional, or inside a longer expression;
the prompt combines outputs such as $2f(3)-g(3)$ ;
one function is placed inside another; or
the same function definition is used more than once.

Use hand substitution when…

the rule and input are simple. For $f(x)=3x+1$ , the calculation $f(4)=3(4)+1=13$ takes only one line.

A question asking for $f(4)$ gives the input and asks for the output. A question asking, “For what value of $x$ is $f(x)=13$ ?” gives the output and asks for the input, so use the one-variable equation workflow.

Look for wording such as:

“What is the value of $f(5)$ ?”
“Which choice is the value of $p(2)+q(2)$ ?”
“A new function is defined by $r(x)=2p(x)-q(x)$ .”
“What is the value of $f(g(4))$ ?”

Check your understanding:

Which is a stronger reason to begin with Desmos: evaluating $f(4)$ for $f(x)=3x+1$ , or evaluating $h(j(-3))$ for two longer function rules? Explain the difference.

1. Define once, then evaluate

In $f(a)=b$ , $a$ is the input and $b$ is the output. To evaluate a function in Desmos:

enter the definition, such as f(x)=x^2-4x+7;
move to a new expression line; and
enter the requested value, such as f(-2).

For $f(x)=x^2-4x+7$ , Desmos gives $f(-2)=19$ . By hand, the same substitution is

f(-2)=(-2)^2-4(-2)+7=4+8+7=19.

Try it yourself:

Before entering f(2), predict whether its output will be greater than or less than $f(-2)$ . Then evaluate it to check.

Check your understanding:

For $f(x)=x^2-4x+7$ , what is $f(2)$ ?

Common mistake:

Treating a negative input as $-2^2$ . The square in $f(-2)$ applies to the entire input, so the substituted term is $(-2)^2$ .

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The number inside the parentheses is the input; Desmos returns the output.

2. Combine outputs without rewriting the rules

Once functions are defined, Desmos can use them in a longer expression. Suppose

p(x)=2x+3

and

q(x)=x^2-1.

To find $p(4)+q(4)$ , enter that expression on a new line. Desmos calculates both outputs and adds them:

p(4)=11,\qquad q(4)=15,

p(4)+q(4)=26.

If the problem defines a new function, enter that definition too. For example, $r(x)=2p(x)-q(x)$ can be entered exactly as written after defining $p$ and $q$ .

Check your understanding:

Using the functions above, a new function is defined by $r(x)=2p(x)-q(x)$ . What is $r(3)$ ?

Common mistake:

Re-entering a function from memory and changing a sign or exponent. Define each rule once, then call the function by name.

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Keep each definition intact and enter the requested combination on a new line.

3. Example: Compose functions from the inside out

A composition uses the output of one function as the input of another. In $h(j(5))$ , evaluate the inside function $j(5)$ first, then use that output as the input of $h$ .

Worked example

The functions $h$ and $j$ are defined by

h(x)=x^2+2x

and

j(x)=3-x

What is the value of

$h(j(5))$ ?

A
$-8$
B
$0$
C
$-2$
D
$4$

Step 1

Read the order from the inside out

The inner calculation is $j(5)$ . Its output becomes the input of $h$ .

Step 2

Define both functions

Enter h(x)=x^2+2x and j(x)=3-x, then enter h(j(5)). Desmos returns $0$ .

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Nested parentheses preserve the inside-out order of the composition.

Step 3

Verify the two stages

First,

j(5)=3-5=-2.

Then,

h(-2)=(-2)^2+2(-2)=4-4=0.

Step 4

Choose the matching answer

The value of $h(j(5))$ is $0$ , so the answer is B.

Common mistake:

Reversing the order and calculating $j(h(5))$ . The outer function is the one written on the outside.

Composition here means evaluating a nested function value. This lesson does not ask you to analyze how a formula transforms an entire graph.

4. Interpret the input, output, and unit

In a contextual function, the input and output represent real quantities. Suppose $d(t)=55t$ gives the distance $d(t)$ , in miles, traveled after $t$ hours. Then $d(3)=165$ means that after $3$ hours, the distance traveled is $165$ miles.

Desmos accepts the function name and input variable from the prompt, so you can enter d(t)=55t as written.

Check your understanding:

The function $A(t)=80(0.75)^t$ gives the amount of medication, in milligrams, remaining $t$ hours after a dose. What does $A(2)$ mean, and what is its value?

Common mistake:

Reporting a calculator output without its meaning. In context, state what the input represents and include the output unit.

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The input is time in hours, and the output is distance in miles.

Finish the solution

The two starting functions are already defined. Define the new function, evaluate it, then submit its value.

Define, then evaluate

Finish the solution

The functions $r$ and $s$ are defined by

r(x)=x^2-2

and

s(x)=x+4.

A new function $v$ is defined by

v(x)=2r(x)-s(x).

What is the value of $v(3)$ ?

First steps

The definitions of $r$ and $s$ are entered.
Enter the definition of $v$ exactly as written.
Evaluate $v(3)$ on a new line.

Finish it

Your answer

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Define v(x) = 2r(x) - s(x), then evaluate v(3).

Practice problems

SAT practice problems

Define each function exactly as written. Then enter the requested value, combination, or composition on a new line.

Evaluate a negative input

Practice problem

The function $f$ is defined by

f(x)=\frac{7-2x}{3}.

What is the value of $f(-1)$ ?

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Combine different outputs

Practice problem

The functions $p$ and $q$ are defined by

p(x)=3x-2

and

q(x)=x^2+4.

What is the value of $p(5)+q(2)$ ?

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Compose functions in context

Practice problem

A factory defines

p(h)=18h+12

as the number of parts produced after $h$ hours. The parts are packed into boxes that hold $6$ parts each. For a number of parts $n$ that fills the boxes exactly, the function

b(n)=\frac{n}{6}

gives the number of full boxes filled by $n$ parts.

What is the value of $b(p(3))$ ?

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Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Use hand substitution for one short calculation; use Desmos when the input or requested expression is longer.
Define each function once, then enter the requested value or combination exactly as written.
If the problem defines a new function from existing functions, enter that definition before evaluating it.
In a composition, evaluate the innermost function first and preserve every set of parentheses.
In context, identify what the input and output represent and report the requested unit.

Next lesson

Inequalities and shaded overlap

Graph inequalities and identify the shaded region that satisfies their conditions.

Start next lesson

Free Practice

What you’ll learn

Prerequisites

Turn function notation into a direct calculation

Fast Desmos solution

When should you evaluate a function in Desmos?

Begin with Desmos when…

Use hand substitution when…

1. Define once, then evaluate

2. Combine outputs without rewriting the rules

3. Example: Compose functions from the inside out

Read the order from the inside out

Define both functions

Verify the two stages

Choose the matching answer

4. Interpret the input, output, and unit

Finish the solution

Define, then evaluate

First steps

Finish it

Practice problems

SAT practice problems

Evaluate a negative input

Combine different outputs

Compose functions in context

4 practice examples left

Quick recap

Inequalities and shaded overlap

Free question bank practice problems