Function Composition Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Function Composition.

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

Concept Recap

Function composition applies one function to the output of another: $(f \circ g)(x) = f(g(x))$ , meaning evaluate $g$ first, then apply $f$ to the result.

Chain two machines together—output of the first goes into the second.

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: Composition feeds the result of the inner function straight into the outer function.

Common stuck point: The procedure for function composition is the easy part; the trap is applying $f$ before $g$ in $(f\circ g)(x)$ . Asking "Is the output of one function being used as the input of another, in a fixed order?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the output of one function being used as the input of another, in a fixed order?

Worked Examples

Example 1

easy

Given

f(x) = 2x + 1

and

g(x) = x^2

, find

(f \circ g)(3)

Answer

19

First step

(f \circ g)(3)

means

f(g(3))

, so evaluate the inner function first.

Full solution

2
Compute $g(3) = 3^2 = 9$ .
3
Substitute that result into $f$ : $f(9) = 2(9) + 1 = 19$ .

Function composition works from the inside out: evaluate the inner function first, then feed its output into the outer function.

Example 2

medium

Given

f(x) = x + 3

and

g(x) = 2x^2 - 1

, find the formula for

(g \circ f)(x)

Example 3

easy

Given

f(x)=x^2

and

g(x)=x-3

, compute both

(f\circ g)(5)

and

(g\circ f)(5)

and notice the order matters.

Example 4

medium

Given

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

and

g(x)=x^2

, find

(f\circ g)(x)

and state its domain.

Example 5

medium

Express

h(x)=\sqrt{2x+5}

as a composition

f\circ g

with simple

f,g

Example 6

hard

Let

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

. Compute

f(f(x))

and

f(f(f(x)))

and describe what you observe.

Example 7

challenge

Define

f_1(x)=\frac{x}{x+1}

and

f_{n+1}=f_1\circ f_n

. Find a closed form for

f_n(x)

Practice Problems

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

Example 1

easy

Given

f(x) = x - 4

and

g(x) = 3x

, find

(f \circ g)(5)

Example 2

hard

Given

f(x) = \sqrt{x}

and

g(x) = x^2 + 5

, find the domain of

(f \circ g)(x)

Example 3

easy

f(x)=x+1

and

g(x)=2x

, find

(f\circ g)(3)

Example 4

easy

f(x)=x^2

and

g(x)=x+3

, find

(f\circ g)(1)

Example 5

easy

f(x)=2x

and

g(x)=x-1

, find

(g\circ f)(4)

Example 6

easy

f(x)=x+2

and

g(x)=3x

, find

(f\circ g)(x)

Example 7

easy

f(x)=x-5

and

g(x)=x+5

, find

(f\circ g)(x)

Example 8

easy

f(x)=x^2

, find

(f\circ f)(2)

Example 9

easy

f(x)=\sqrt{x}

and

g(x)=x+9

, find

(f\circ g)(0)

Example 10

easy

f(x)=3x+1

and

g(x)=x

, find

(f\circ g)(x)

Example 11

medium

f(x)=x+1

and

g(x)=x^2

, find

(f\circ g)(x)

and

(g\circ f)(x)

Example 12

medium

f(x)=2x-1

and

g(x)=\frac{x+1}{2}

, find

(f\circ g)(x)

Example 13

medium

h(x)=(x+3)^2

, write

h

f\circ g

Example 14

medium

f(x)=x^2+1

and

(f\circ g)(x)=x^2+2x+2

, find

g(x)

(linear).

Example 15

medium

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

and

g(x)=x-2

, find the domain of

(f\circ g)(x)

Example 16

medium

f(x)=3x

, find a function

g

with

(f\circ g)(x)=x

Example 17

medium

f(x)=x+a

and

(f\circ f)(x)=x+6

, find

a

Example 18

medium

f(x)=2x

and

g(x)=x+1

, find

(f\circ g\circ f)(2)

Example 19

challenge

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

, find

(f\circ f)(x)

simplified.

Example 20

challenge

(f\circ g)(x)=4x^2+4x+2

and

f(x)=x^2+1

, find a linear

g(x)

Example 21

challenge

Functions satisfy

f(x)=2x+3

and

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

for all

x

, with

g

linear

g(x)=mx+b

. Find a relation between

m

and

b

Example 22

medium

f(x)=x-4

and

g(x)=x^2

, find

(g\circ f)(x)

Example 23

easy

f(x)=4x-3

and

g(x)=x+2

, find

(f\circ g)(5)

Example 24

easy

f(x)=x^2-1

and

g(x)=2x

, find

(f\circ g)(3)

Example 25

easy

f(x)=x+7

and

g(x)=2x-1

, find

(f\circ g)(x)

Example 26

medium

f(x)=2x+5

and

g(x)=x^2-4

, find

(f\circ g)(x)

Example 27

medium

f(x)=x^2+x

and

g(x)=x-1

, find

(f\circ g)(x)

simplified.

Example 28

medium

f(x)=\frac{1}{x-3}

and

g(x)=x+1

, find the domain of

(f\circ g)(x)

Example 29

medium

f(x)=2x+1

and

(f\circ g)(x)=2x+7

, find a linear

g(x)

Example 30

medium

g(x)=x-2

and

(f\circ g)(x)=x^2-4x+1

, find

f(x)

Example 31

medium

f(x)=3x-1

, find

(f\circ f)(x)

Example 32

medium

Express

h(x)=(3x-7)^4

f\circ g

Example 33

medium

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

, find

(f\circ f)(x)

for

x\ne 0

Example 34

hard

Given

f(x)=x^2+1

and

(g\circ f)(x)=2x^2+5

, find

g(x)

Example 35

hard

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

, find

(f\circ f)(x)

in simplified form for

x\ne 1

Example 36

hard

f(x)=ax+b

and

(f\circ f)(x)=9x+8

, find all real

(a,b)

Example 37

hard

Find a function

g

with

(g\circ g)(x)=4x+3

, assuming

g

is linear.

Example 38

hard

f(x)=2x+1

and

g(x)=x-3

, find

(f\circ g)^{-1}(x)

Example 39

challenge

Find all linear functions

f(x)=ax+b

(with

a\ne 0

) such that

f\circ f= f^{-1}

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Related Concepts

Function Inverse Function

Background Knowledge

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

function definition