Input-Output View Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Input-Output View.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The input-output view of a function treats it as a black box: put in a value (input), get out a uniquely determined value (output), without worrying about the internal mechanism.

Like a vending machine: put in selection (input), get out snack (output).

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The input-output view treats a function as a machine: a value goes in and one determined value comes out, ignoring the inner workings.

Common stuck point: The procedure for input-output view is the easy part; the trap is letting one input map to two outputs in the box. Asking "Am I tracking what comes out for a given input, treating the rule as a sealed box?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I tracking what comes out for a given input, treating the rule as a sealed box?

Worked Examples

Example 1

easy

Think of

f(x) = 3x - 7

as a machine. Describe the sequence of operations applied to input

x

, then evaluate

f(5)

and find the input that gives output

14

Answer

f(5)=8

; input

x=7

gives output

14

First step

Machine description: take input

x

→ multiply by

3

→ subtract

7

→ output.

Full solution

2
Evaluate: $f(5) = 3(5)-7 = 15-7 = 8$ .
3
Find input for output $14$ : solve $3x-7=14 \Rightarrow 3x=21 \Rightarrow x=7$ .

The input-output view treats a function as a process or machine. This perspective makes it natural to evaluate forward (given input, find output) and backward (given output, find input), building intuition for inverse operations.

Example 2

medium

A function machine applies two operations in sequence: first square the input, then add

3

. Write the function formula

f(x)

, fill in a table for

x \in \{-2,-1,0,1,2\}

, and identify any symmetry.

Example 3

medium

For

f(x) = x^2 - x

, compute

f(3) - f(2)

Example 4

medium

f(x) = ax + b

and

f(2) = 5

f(4) = 11

, find

a

and

b

Example 5

hard

For

f(x) = x^2

, evaluate

\dfrac{f(x + h) - f(x)}{h}

and simplify.

Example 6

challenge

Given

f(x) = 2x + 3

, find a function

g

so that

f(g(x)) = x

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A vending machine charges

\$1.50

per item. Write a function

C(n)

for the cost of

n

items, and find

C(4)

and the number of items for budget

\$9

Example 2

medium

Two function machines are connected: machine

A

doubles the input, machine

B

subtracts

1

. Find the combined output for input

x=5

, and write the combined formula.

Example 3

easy

f(x)=x+5

, what is the output when the input is

3

Example 4

easy

Does

f(x)

mean

f

multiplied by

x

Example 5

easy

A machine adds 2 then doubles. What is the output for input

3

Example 6

easy

f(7)=12

, which number is the input and which is the output?

Example 7

easy

g

squares its input, what is

g(4)

Example 8

easy

f

the same as

f(3)

Example 9

easy

Must the inputs of a function be numbers?

Example 10

easy

A function takes input

x

and outputs

x-1

. What is the output for input

0

Example 11

medium

For

f(x)=3x-2

, find the input that produces output

10

Example 12

medium

Two machines:

f

doubles,

g

adds 3. Find

g(f(2))

Example 13

medium

For the same machines, find

f(g(2))

and compare to

g(f(2))

Example 14

medium

f(x)=x^2

, evaluate

f(a+1)

Example 15

medium

A function machine outputs

2x+1

. If two inputs give outputs

5

and

9

, find the inputs.

Example 16

medium

f(3)=7

and

f(5)=7

, is

f

a function? Is it one-to-one?

Example 17

medium

A temperature converter takes Celsius input

C

and outputs

F=\tfrac{9}{5}C+32

. Find the output for input

C=20

Example 18

medium

f(x)=\dfrac{1}{x-2}

, which input is NOT allowed, and why (input-output view)?

Example 19

medium

f(x)=x^2+1

, find

f(f(1))

Example 20

challenge

Machines:

f(x)=2x

and

g(x)=x+3

. Find an input

x

with

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

Example 21

challenge

A machine doubles then subtracts 1:

f(x)=2x-1

. Find the rule that reverses it (the inverse machine).

Example 22

challenge

A function table shows inputs

1,2,3

giving outputs

2,4,8

. Find a rule

f(n)

matching it.

Example 23

easy

f(x) = 2x + 1

, what is

f(6)

Example 24

easy

A machine triples the input. What is the output for input

-4

Example 25

easy

g

subtracts

5

from its input, what input gives output

0

Example 26

easy

For

f(x) = x + 4

, fill in the table:

f(0), f(1), f(-2)

Example 27

easy

A function maps each student to their birth year. Is birth year the input or the output?

Example 28

medium

For

f(x) = 4x - 9

, find the input that gives output

-1

Example 29

medium

f(x) = 2x + 1

and

g(x) = x - 3

, find

f(g(5))

Example 30

medium

f(x) = 2x + 1

and

g(x) = x - 3

, find

g(f(5))

and compare to

f(g(5))

Example 31

medium

For

h(x) = \frac{x + 2}{x - 1}

, find

h(3)

Example 32

medium

For

f(x) = 3 - 2x

, find

f(a + 1)

Example 33

medium

A function table has inputs

1, 2, 3, 4

giving outputs

3, 5, 7, 9

. Write a formula.

Example 34

hard

f(x) = 3x + 1

, find the inverse machine that recovers

x

from

f(x)

Example 35

hard

For

f(x) = x^2 + 1

, find

f(2t) - 4f(t)

as an expression in

t

Example 36

hard

f(x) = \frac{1}{x}

and

g(x) = x - 2

, find the domain of

f \circ g

Example 37

hard

For

f(x) = \dfrac{x + 1}{x - 1}

, find

f(f(2))

Example 38

challenge

For

f(x) = \frac{1}{1 - x}

, compute

f(f(f(x)))

and simplify.

Example 39

challenge

A function

f

satisfies

f(x) + 2 f(1 - x) = x

for all

x

. Find

f(0)

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Related Concepts

Function Function Composition Function Families

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition