Evaluate, combine, and compose functions

Your feedback matters:

Lesson progressPractice problems 0/4

Define SAT functions in Desmos, evaluate direct and combined outputs, compose functions in the correct order, and interpret results in context.

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

What you’ll learn

Tell the input from the output in function notation.
Define a function once and evaluate it at any input.
Work out expressions that combine outputs, like $2f(3)-g(3)$ .
Define and evaluate a new function built from other functions.
Compose functions from the inside out.
Say what a function value means, with the right unit.
Spot when a quick substitution by hand beats opening Desmos.

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 hide the number they want inside notation like $f(4)$ or $2f(3)-g(3)$ . Harder ones put one function inside another. Desmos reads all of these exactly as they're written, so you don't have to untangle them first.

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 $2f(-2)-g(-2)$ ?

Fast Desmos solution

Type the two rules on separate lines:

f(x)=x^2+3

and

g(x)=4x-1.

Then type 2f(-2)-g(-2) on a third line. Desmos shows $23$ , so the answer is D.

Why $23$ and not $5$ ? Here $f(-2)=7$ and $g(-2)=-9$ , and subtracting $-9$ adds $9$ , so you get $14+9=23$ . Desmos keeps track of signs like that for you. Your job is to pick out what the question wants and to know what the answer means.

Calculator loads as you approach

Desmos finds both outputs and combines them in one line.

When should you evaluate a function in Desmos?

Desmos earns its keep when one substitution grows into a longer calculation: a long expression, one function inside another, or the same rule used again and again.

Start with Desmos

You’re given a function rule and asked for one or more outputs, like $f(5)$ .
The input is awkward, like $-2$ or $\frac{1}{2}$ , or it sits inside a longer expression.
The question combines outputs, like $2f(3)-g(3)$ .
One function sits inside another, like $h(j(5))$ .
You’ll need the same rule more than once, so typing it once saves you from copying it again.

Work it by hand

The rule and the input are both simple. For $f(x)=3x+1$ , you get $f(4)=3(4)+1=13$ in one line, faster than typing anything.

Before either one, check which way the question runs. $f(4)$ gives you the input and asks for the output. “For what value of $x$ is $f(x)=13$ ?” gives you the output and asks for the input, so that’s an equation to solve with the one-variable equation workflow.

You'll recognize these questions by wording like this:

“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 would you start in Desmos: $f(4)$ for $f(x)=3x+1$ , or $h(j(-3))$ for two longer rules? Why?

Define once, then evaluate

f(a)=b,

$a$ is the input, the number you put in, and $b$ is the output, the number the function gives back. To evaluate a function in Desmos:

Type the rule, like f(x)=x^2-4x+7.
Go to a new line.
Type the value you want, like f(-2).

For

f(x)=x^2-4x+7,

Desmos shows

f(-2)=19.

Here's the same substitution by hand:

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

Once $f$ is defined, any other input is one more short line, like f(2) or f(10). That's the habit to build: define it once, then call it by name.

Try it yourself:

Before you type f(2), make a guess: will the output be bigger or smaller than $f(-2)$ ? Then type it to check.

Check your understanding:

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

Common mistake:

Writing the squared term of $f(-2)$ as $-2^2$ , which is $-4$ . The whole input gets squared, so the term is $(-2)^2=4$ . When you substitute by hand, put the input in parentheses first.

Calculator loads as you approach

The number in the parentheses is the input. Desmos gives back the output.

Combine outputs without retyping the rules

Once your functions are defined, you can use their names inside a longer expression. Say

p(x)=2x+3

and

q(x)=x^2-1.

To find

p(4)+q(4),

type exactly that on a new line. Desmos works out both outputs and adds them:

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

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

Sometimes a question builds a new function out of old ones, like

r(x)=2p(x)-q(x).

After $p$ and $q$ are in, type that definition exactly as it's written. From then on, $r$ works like any other function.

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:

Retyping a rule from memory and slipping on a sign or an exponent, like writing $x^2+1$ for $q$ . Every answer that uses it is off. Type each rule once and use its name after that.

Calculator loads as you approach

Leave each rule as it is and type the combination on a new line.

Example: Compose functions from the inside out

A composition puts one function inside another, so the output of one becomes the input of the next. In

h(j(5)),

you work from the inside out: find $j(5)$ first, then feed that number into $h$ . Inside first, outside last.

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

Find the inside piece

The inside piece is $j(5)$ . Whatever it gives you becomes the input of $h$ .

Step 2

Define both functions

Type h(x)=x^2+2x and j(x)=3-x, then type h(j(5)). Desmos shows $0$ .

Calculator loads as you approach

The nested parentheses keep the order: inside first, then outside.

Step 3

Check the two stages

First,

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

That $-2$ goes into $h$ :

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

Step 4

Choose the matching answer

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

Common mistake: Working in the wrong order and finding $j(h(5))$ instead. That gives $j(35)=-32$ , a completely different number. The function written on the outside is the one you use last.

Interpret the input, output, and unit

In a word problem, the input and output are real quantities, with units. Say

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 is fine with the function name and letter the question uses, so you can type d(t)=55t exactly as it's written.

Check your understanding:

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

Common mistake:

Giving a calculator output, like $165$ , without its meaning. In a word problem, say what the input stands for and give the output’s unit: after $3$ hours, $165$ miles.

Calculator loads as you approach

The input is time in hours. The output is distance in miles.

Finish the solution

Here, $v$ is built from $r$ and $s$ . Define it, find $v(3)$ , and enter that value.

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 rules for $r$ and $s$ are already in.
Type the definition of $v$ exactly as it's written.
Find $v(3)$ on a new line.

Finish it

Your answer

Calculator loads as you approach

Add v(x) = 2r(x) - s(x) below r and s, then evaluate v(3).

Practice problems

Your turn: one single value, one sum of outputs and one composition.

Practice problem

The function $f$ is defined by

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

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

Calculator loads as you approach

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 $p(5)+q(2)$ ?

Calculator loads as you approach

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 $b(p(3))$ ?

Calculator loads as you approach

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Do a short substitution by hand. Use Desmos when the input or the expression gets longer.
Define each function once, then type the value or combination exactly as the question writes it.
When a question builds a new function from old ones, type its definition before you evaluate it.
In a composition, work inside first, and keep every set of parentheses.
In a word problem, say what the input and output mean, and give the unit the question asks for.

Next lesson

Inequalities and shaded overlap

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

Start next lesson

Practice

Practice this lesson

417 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

Start practice