Find inputs and outputs of linear functions

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Lesson progressPractice problems 0/3

Difficulty: Beginner
Estimated time: 28 minutes
Domains: Algebra
Techniques: Function-notationFunction-evaluationReverse-input-recovery

What you’ll learn

Tell which number is the input and which is the output in function notation.
Find the output for a given input.
Find the input that gives a given output.
Tell when to type the work straight into Desmos: when the decimals or negative signs aren't quick in your head.
Say what a function value means, with the right quantity and units.

Prerequisites

Why this matters on the SAT

Find the missing side first

A function is a rule that turns an input into an output. SAT questions come at it from both ends. Some give you the input and ask for the output. Others give you the output and ask which input produced it. The arithmetic is usually short. The step that matters is the first one: figure out which side is missing, so you know which way to go.

Going backward means solving a linear equation. The rule itself will look like the ones you worked with in writing linear functions, except here it's already written for you.

Once you know the direction, look at the numbers to decide how to do the work:

When the input or rule has decimals or negative signs that aren't quick in your head, like $-3(-4)+11$ , type the whole substituted expression into Desmos on one line.
When going backward leaves a decimal equation that isn't quick to solve, like the one in the example below, type that equation into Desmos exactly as it stands.
When it's a quick exact substitution or an equation you can solve in one or two steps, do it by hand. Matching an expression inside the parentheses, like $3x-2$ in $p(3x-2)$ , to the input you want is quicker by hand too.

SAT example

A bicycle rental company uses the function

C(h)=18+6.5h

to calculate the total cost $C(h)$ , in dollars, of renting a bicycle for $h$ hours. If $C(a)=70$ , what is the value of $a$ ?

A
$6.5$
B
$7$
C
$8$
D
$52$

Solution to the example

The question gives you the output, $70$ dollars, and asks for the input. So put the rule in place of $C(a)$ and set it equal to $70$ :

18+6.5a=70

That decimal makes it slow to solve in your head, so hand it to Desmos. Desmos graphs with lowercase $x$ , so type 18+6.5x=70 with $x$ in place of $a$ . Click the vertical line that appears to read $x=8$ . That's $a=8$ in the problem.

The answer is C. In words, $C(8)=70$ means an $8$ -hour rental costs $70.

Calculator loads as you approach

Click the vertical line to read

x=8

. In the problem, that means

a=8

hours.

Spot the input and the output

A function matches each allowed input with exactly one output. When you see

f(a)=b,

read it as "put in $a$ , get out $b$ ." So $a$ is the input and $b$ is the output. It also tells you that the point $(a,b)$ is on the graph of $y=f(x)$ , with the input first.

In a rule like

f(x)=3x-5,

the $x$ is a placeholder. It marks the spot where whatever input you're given goes into the rule. The letters can change, too. In $T(t)$ , the function is named $T$ and its input is $t$ .

Which way do you go?

What the question gives	What’s missing	What you do
$f(4)$	The output	Put $4$ in for the input and work it out.
$f(a)=17$	The input $a$	Set the rule equal to $17$ and solve.
A table row $(3,11)$	Nothing, the pair is complete	Read it as $f(3)=11$ .

Here's a short way to hold on to it: inside the parentheses goes in, after the equals sign comes out. Whichever one the question leaves blank is the one you're finding.

Selected values of $p$

Input $x$	Output $p(x)$
$-3$	$14$
$0$	$8$
$2$	$4$

Check your understanding:

The table shows some values of $p$ . Which row shows $p(a)=4$ ? What is $a$ , and what point does that row give on the graph of $y=p(x)$ ?

Go forward when you know the input

To evaluate a function, put the input in everywhere the input variable appears, then do the arithmetic.

Say

q(x)=-3x+11,

and you want $q(-4)$ . Put $-4$ in for $x$ . The signs make this easy to fumble in your head, so type the whole substituted expression into Desmos:

-3(-4)+11

Desmos returns $23$ , so $q(-4)=23$ . You don't need to define q(x) in Desmos first for one value like this. Defining the function pays off only when you'll evaluate it more than once or use it again later.

The input won't always be a number. If it's an expression, put in the whole expression, parentheses and all:

\begin{aligned} q(t+2) &=-3(t+2)+11\\[1.4em] &=-3t-6+11\\[1.4em] &=-3t+5. \end{aligned}

Stop simplifying once you reach the form the question asks for.

Common mistake:

A common slip is dropping the negative sign: $-3(4)+11=-1$ , when $q(-4)$ is really $-3(-4)+11=23$ . The input is $-4$ , not $4$ , so write it in parentheses every time you substitute. Then check that the sign makes sense. Here $q$ goes down as $x$ goes up, so a negative input should give an output above $11$ .

Check your understanding:

If $r(t)=\frac{7-2t}{3}$ , what is $r(-1)$ ? Write the substituted expression before you simplify.

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Try deleting the parentheses around

-4

and watch the output change. Then change the input and see what comes out.

Go backward when you know the output

Now flip it around. When the question says

f(a)=k,

you know the output, $k$ , and the input $a$ is what you're after. Write the rule with $a$ as the input, set it equal to $k$ , and solve.

Say

f(x)=4x-7

and $f(a)=29$ . Write

4a-7=29,

then solve:

\begin{aligned} 4a-7&=29\\[1.4em] 4a&=36\\[1.4em] a&=9. \end{aligned}

Check it by going forward:

f(9)=4(9)-7=29.

A linear function that isn't constant sends each output back to exactly one input, so you'll find one answer. The exception is a horizontal line. If $f(x)=7$ , every input gives $7$ , and no input gives $10$ .

Common mistake:

When the question says $f(a)=29$ , it’s tempting to work out $f(29)$ . That treats the output as an input. Look at where things sit: $a$ is inside the parentheses, so it’s the unknown input, and $29$ is the output. Set the rule equal to $29$ , solve for $a$ , and plug your answer back in to check.

Check your understanding:

The function $s$ is defined by $s(t)=7-2t$ . If $s(k)=-9$ , what is $k$ ? How can you check it?

Read the pair in context

In a word problem, $f(a)=b$ is still an input and an output. The story tells you what each one counts and in what units.

Say

M(w)=64+8w

gives the number of members $M(w)$ in a club $w$ weeks after registration opens. Then

M(5)=104

says three things:

The input is $5$ weeks after registration opens.
The output is $104$ members.
Put together: after $5$ weeks, the club has $104$ members.

Don't flip it. It doesn't say that $104$ weeks give $5$ members.

So when you find an input, give it the input's unit, and when you find an output, give it the output's unit. The right number in the wrong role, or with the wrong unit, doesn't answer the question.

Check your understanding:

The function $H(t)=180-4t$ gives a drone’s altitude $H(t)$ , in meters, $t$ seconds after it starts descending. What does $H(16)=116$ mean? Say it in one full sentence.

Example: Let the input change do the work

Worked example

The linear function $f$ is defined by

f(x)=\frac52x-9.

If $f(a)=26$ , what is the value of $f(a-4)$ ?

A
$6$
B
$10$
C
$16$
D
$26$

Step 1

Use the slope-change shortcut

You might expect to find $a$ first. You don't have to. In a linear function, the output changes by the slope times the input change, no matter where the input starts. The slope here is $\frac52$ , and the input moves from $a$ to $a-4$ , a change of $-4$ . So the output changes by

\frac52(-4)=-10.

Start from the output you know, $f(a)=26$ , and apply that change:

26+(-10)=16.

So $f(a-4)=16$ , and the answer is C. You never needed the value of $a$ .

Step 2

Or find the input first

Finding the input first works too, and it's the general method to fall back on when no handy slope change shows up. Put the rule in place of $f(a)$ and solve:

\begin{aligned} \frac52a-9&=26\\[1.4em] \frac52a&=35\\[1.4em] a&=14. \end{aligned}

The input you want is $a-4$ , so

a-4=14-4=10.

Step 3

Finish the other way

Evaluate $f$ at that new input:

\begin{aligned} f(10) &=\frac52(10)-9\\[1.4em] &=25-9\\[1.4em] &=16. \end{aligned}

Same answer, C.

Check your understanding:

You found $a=14$ along the way. Why isn’t $14$ the answer?

Practice problems

Your turn. For each one, ask first: is the input missing, or the output?

Evaluate a negative input

Practice problem

The function $g$ is defined by

g(x)=\frac{3-5x}{4}.

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

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Type the whole substituted expression here as one line.

Find a time from an amount

Practice problem

The function

A(t)=950-32t

gives the amount of water $A(t)$ , in liters, in a tank $t$ minutes after draining begins. After how many minutes will the tank contain $566$ liters of water?

Your answer

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Type the equation with

x

in place of

t

, then click its vertical line.

Match an expression input

Practice problem

The function $p$ is linear. For all real values of $x$ ,

p(3x-2)=12x+7.

What is the value of $p(13)$ ?

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Match

3x-2

13

by hand first. Use this space to check if you like.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

In $f(a)=b$ , $a$ is the input and $b$ is the output, and $(a,b)$ is a point on the graph.
Know the input? Put the whole input into the rule and work forward.
Know the output? Set the rule equal to it, solve for the input, and check by going forward.
Do quick, exact work by hand. Type messy decimals or signs into Desmos as one expression or one equation, and define the function only when you'll reuse it.
In context, the input carries the input's unit and the output carries the output's unit.
For a statement like $p(3x-2)=12x+7$ , set the whole inner expression equal to the input you want before you read off the output.

Related lessons

No rule yet, only points, a table or a graph? Find the rule first with Write and match linear functions.
Rules that aren't linear can send more than one input to the same output. Evaluate nonlinear functions and recover inputs shows you how to handle them.

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Recognize a contextual constant rate, build the linear rule, and use it for a prediction.

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