Composition Chains Examples in Math

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

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

Concept Recap

A composition chain is a sequence of functions applied one after another: $(f \circ g \circ h)(x) = f(g(h(x)))$ , evaluated inside-out from right to left.

Work from the innermost function outward — compute $h(x)$ first, then feed that result to $g$ , then feed that to $f$ . The order matters critically.

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: A composition chain feeds one function's output into the next, evaluated from the innermost function outward.

Common stuck point: The procedure for composition chains is the easy part; the trap is evaluating left to right. Asking "Is one function's output being fed as the input to the next 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 one function's output being fed as the input to the next in a fixed order?

Worked Examples

Example 1

easy

Let

f(x)=x+1

g(x)=2x

h(x)=x^2

. Compute

(f\circ g\circ h)(3)

step by step.

Answer

(f\circ g\circ h)(3) = 19

First step

Start from the innermost function:

h(3)=3^2=9

Full solution

2
Apply $g$ : $g(h(3))=g(9)=2(9)=18$ .
3
Apply $f$ : $f(g(h(3)))=f(18)=18+1=19$ . So $(f\circ g\circ h)(3)=19$ .

Function composition chains are evaluated from right to left (innermost to outermost). The output of each function becomes the input of the next. Keeping track of this order is essential for correct computation.

Example 2

medium

Find the formula for

(g\circ f)(x)

and

(f\circ g)(x)

where

f(x)=x^2+1

and

g(x)=\sqrt{x}

. Show they are not equal.

Example 3

medium

Let

f(x) = x^2 - 1

and

g(x) = \sqrt{x + 1}

. Find

(f \circ g)(x)

and state where it is defined.

Example 4

medium

Given

f(x) = e^x

g(x) = 2x

h(x) = x + 3

, find

(f \circ g \circ h)(x)

Example 5

medium

Decompose

H(x) = (x + 4)^5

into two simpler functions

f

and

g

so that

H = f \circ g

Example 6

hard

Decompose

H(x) = \sqrt{\sin(3x)}

into three simpler functions.

Example 7

hard

Given

f(x) = x^2 + 1

and

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

, find

g(x)

Example 8

hard

Let

f(x) = x + a

and

g(x) = bx

. Find

a

and

b

so that

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

and

(g \circ f)(x) = 2x + 10

Example 9

challenge

Let

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

defined for

x \neq -1

. Find

f(f(f(x)))

in simplest form.

Practice Problems

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

Example 1

easy

Given

f(x)=3x-1

and

g(x)=x^2

, find

(f\circ g)(x)

and evaluate it at

x=2

Example 2

hard

Decompose

H(x)=\sin(e^{x^2})

as a composition

H=f\circ g\circ h

of three simpler functions.

Example 3

easy

f(x) = x + 1

and

g(x) = 2x

, find

(f \circ g)(3) = f(g(3))

Example 4

easy

g(x) = x^2

and

f(x) = x - 5

, find

(g \circ f)(7) = g(f(7))

Example 5

easy

With

h(x) = x + 2

g(x) = 3x

f(x) = x - 1

, find

(f \circ g \circ h)(1)

Example 6

easy

f(x) = x + 4

and

g(x) = x - 4

, find

(f \circ g)(x)

Example 7

easy

f(x) = 2x

and

g(x) = x + 3

, write the formula for

f(g(x))

Example 8

easy

f(x) = 2x

and

g(x) = x + 3

, write the formula for

g(f(x))

and note it differs from

f(g(x))

Example 9

easy

Given the chain

f(g(h(x)))

, which function is applied to

x

FIRST?

Example 10

easy

f(x) = \sqrt{x}

and

g(x) = x + 9

, find

(f \circ g)(7)

Example 11

medium

With

f(x) = x^2

and

g(x) = x - 3

, write both

(f \circ g)(x)

and

(g \circ f)(x)

and find where they are equal.

Example 12

medium

(f \circ g)(x) = f(g(x))

with

g(x) = x + 1

and

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

, find

f(x)

Example 13

medium

Express

H(x) = \sqrt{3x + 1}

as a composition

f(g(x))

with

f

and

g

as simple as possible.

Example 14

medium

With

f(x) = x + 1

applied repeatedly, find

(f \circ f \circ f)(x)

(apply

f

three times).

Example 15

medium

With

f(x) = 2x

applied repeatedly, find a formula for applying

f

exactly

n

times to

x

Example 16

medium

Functions:

h(x) = x^2

g(x) = x + 1

f(x) = 3x

. Compute

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

Example 17

medium

Two functions:

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

(domain

x \ne 0

) and

g(x) = x - 2

. State the domain restriction needed for

(f \circ g)(x)

Example 18

challenge

Given

f(x) = x + a

and

g(x) = bx

, find conditions on

a

and

b

so that

f \circ g = g \circ f

for all

x

Example 19

challenge

f(f(x)) = x

for all

x

and

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

form

f(x) = c - x

, verify

f

is its own inverse and find

f(f(5))

for

c = 8

Example 20

challenge

Decompose

H(x) = (2x + 5)^3 + 1

into a chain

f(g(h(x)))

of three simple functions and identify the application order.

Example 21

medium

With

f(x) = x^2

and

g(x) = x + 2

, find

(g \circ f)(3)

and

(f \circ g)(3)

to show they differ.

medium

For

f(x) = 1/x

and

g(x) = x - 2

, find

(g \circ f)(x)

and state where it is defined.

Example 29

medium

Let

f(x) = 2x - 1

and

g(x) = (x + 1)/2

. Compute

(f \circ g)(x)

and

(g \circ f)(x)

Example 30

medium

f(x) = x^2

and

g(x) = 3x + 1

, find

(f \circ g)(x)

and

(g \circ f)(x)

as formulas.

Example 31

medium

For

f(x) = \ln(x)

and

g(x) = x^2 + 1

, find the domain of

(f \circ g)(x)

Example 32

medium

Given

f(x) = 2x

, compute

(f \circ f \circ f)(5)

Example 33

hard

f(x) = ax + b

and

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

, find

a

and

b

(with

a > 0

Example 34

hard

For

f(x) = \sqrt{x}

and

g(x) = 4 - x^2

, find the domain of

(f \circ g)(x)

Example 35

hard

For

f(x) = x^2

and

g(x) = x - 3

, find all

x

with

(f \circ g)(x) = (g \circ f)(x)

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Related Concepts

Function Composition Chain Rule Decomposition

Background Knowledge

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

composition