Unit 2: Functions

Topic 2.5: Composite and Inverse Functions Questions

Practice 20 exam-style questions for IB Math AA SL Topic 2.5. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.

1Find2 marks

2026

The graph of

y = f (x)

passes through (1, 5) and (5, 2). Find (a) f(1); (b) f⁻¹(2).

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2Choose1 mark

How is the graph of

y = f^{- 1} (x)

obtained from the graph of

y = f (x)

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3Write down1 mark

2021

The points

(2, 5)

and

(5, - 1)

lie on the curve

y = f (x)

.

State the value obtained by applying

f

twice, starting from

2

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4Write down1 mark

2026

Write down an expression for

(g \circ f) (x)

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5Find2 marks

2026

f(x) = x² and g(x) = 2x − 1. Find (g∘f)(3).

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6Find2 marks

2026

The graph of y = f(x) passes through (1, 5) and (5, 2). Find (a) f(1); (b) f⁻¹(2).

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7Write down2 marks

The graph of a function f passes through the points (−2, 1), (0, 3), (1, 5) and (3, 8), and f is one-to-one. Write down f⁻¹(3), and the coordinates of the point on the graph of f⁻¹ that corresponds to (3, 8).

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8Write down1 mark

2026

Write down

f^{- 1} (x)

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9Find3 marks

The functions f and g are given by the tables of values: f(1) = 4, f(2) = 7, f(3) = 2, f(4) = 5, and g(1) = 3, g(2) = 1, g(3) = 4, g(4) = 2. Find (f ∘ g)(2) and f⁻¹(5).

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10Choose1 mark

For a function

f

f (4) = 9

. Write down

f^{- 1} (9)

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11Find2 marks

2026

f (x) = 3 x - 2

and

g (x) = x + 4

. Find

(f \circ g) (2)

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12Write down1 mark

2023

Let

f (x) = 2 x + 1

. It is given that

f^{2} (x) = 4 x + 3

.

Write down an expression for

f^{3} (x)

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13Find2 marks

2026

Find the inverse of

f (x) = 5 x - 2

, giving your answer as

f^{- 1} (x)

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14Write down1 mark

2021

A function satisfies

f (0) = 3

and

f (3) = - 2

.

Write down the value of

(f \circ f) (0)

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15Show that2 marks

2026

The functions f and g are defined by f(x) = x² + 4x, x ∈ ℝ, and g(x) = ln x − 1, x > 0.

Show that (f ∘ g)(x) = (ln x)² + 2 ln x − 3.

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16Show that2 marks

2026

The functions f and g are defined by f(x) = x² − 3x and g(x) = e^x + 2, x ∈ ℝ.

Show that (f ∘ g)(x) = e^2x + e^x − 2.

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17Find2 marks

2026

f (x) = x^{2}

and

g (x) = 2 x - 1

. Find

(g \circ f) (3)

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18Write down2 marks

2023

Let

f (x) = x - 2

and

g (x) = x^{2} + k

, where

k

is a constant.

Write down

(g \circ f) (x)

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19Find2 marks

2026

f(x) = 3x − 2 and g(x) = x + 4. Find (f∘g)(2).

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20Find2 marks

2026

Find the inverse of f(x) = 5x − 2, giving your answer as f⁻¹(x).

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