Practice Composition Chains in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy

Let

f(x) = 2x + 1

and

g(x) = x - 4

. Write the formula for

(g \circ f)(x)

Example 2

easy

f(x) = 2x

and

g(x) = x + 3

, write the formula for

f(g(x))

Example 3

easy

Given

f(x)=3x-1

and

g(x)=x^2

, find

(f\circ g)(x)

and evaluate it at

x=2

Example 4

medium

Given

f(x) = x + 2

, find

(f \circ f)(x)

Example 5

hard

Decompose

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

as a composition

H=f\circ g\circ h

of three simpler functions.

Example 6

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 7

hard

f(x) = 1/x

for

x \neq 0

, find

(f \circ f)(x)

Example 8

easy

For

f(x) = x + 3

and

g(x) = 4x

, find

g(f(2))

Example 9

medium

With

f(x) = x + 1

applied repeatedly, find

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

(apply

f

three times).

Example 10

easy

Let

f(x)=x+1

g(x)=2x

h(x)=x^2

. Compute

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

step by step.

Example 11

medium

f(x) = x^2

and

g(x) = 3x + 1

, find

(f \circ g)(x)

and

(g \circ f)(x)

as formulas.

Example 12

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 13

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 14

easy

f(x) = x + 4

and

g(x) = x - 4

, find

(f \circ g)(x)

Example 15

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 16

easy

Let

f(x) = x^2

and

g(x) = x + 1

. Find

(g \circ f)(3)

Example 17

medium

Let

f(x) = 2x - 1

and

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

. Compute

(f \circ g)(x)

and

(g \circ f)(x)

Example 18

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 19

medium

Functions:

h(x) = x^2

g(x) = x + 1

f(x) = 3x

. Compute

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

Example 20

easy

For

f(x) = 3x

and

g(x) = x - 6

, write the formula for

f(g(x))

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Related Concepts

Function Composition Chain Rule Decomposition