Practice Function Composition in Math

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

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy

f(x)=x+7

and

g(x)=2x-1

, find

(f\circ g)(x)

Example 2

medium

f(x)=2x+1

and

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

, find a linear

g(x)

Example 3

medium

f(x)=3x

, find a function

g

with

(f\circ g)(x)=x

Example 4

medium

f(x)=x^2+1

and

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

, find

g(x)

(linear).

Example 5

medium

f(x)=2x+5

and

g(x)=x^2-4

, find

(f\circ g)(x)

Example 6

medium

f(x)=x^2+x

and

g(x)=x-1

, find

(f\circ g)(x)

simplified.

Example 7

medium

f(x)=x^2

and

g(x)=x+1

, evaluate

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

Example 8

medium

Express

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

as a composition

f\circ g

with simple

f,g

Example 9

easy

f(x)=x+1

and

g(x)=2x

, find

(f\circ g)(3)

Example 10

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 11

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)

Example 12

hard

Given

f(x) = \sqrt{x}

and

g(x) = x^2 + 5

, find the domain of

(f \circ g)(x)

Example 13

hard

Find a function

g

with

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

, assuming

g

is linear.

Example 14

medium

f(x)=x+a

and

g(x)=x+b

, find

(f\circ g)(x)

Example 15

medium

Express

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

f\circ g

Example 16

medium

f(x)=x+1

and

g(x)=x^2

, find

(f\circ g)(x)

and

(g\circ f)(x)

Example 17

medium

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

, write

h

f\circ g

Example 18

medium

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

and

g(x)=x-2

, find the domain of

(f\circ g)(x)

Example 19

medium

Given

f(x) = x + 3

and

g(x) = 2x^2 - 1

, find the formula for

(g \circ f)(x)

Example 20

easy

f(x)=x^2

and

g(x)=x+3

, find

(f\circ g)(1)

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