IB · Mathematics: Analysis & Approaches SL & HL · Functions

Functions, composites & inverses: IB Mathematics: Analysis & Approaches questions with answers

Free Functions, composites & inverses questions for IB Mathematics: Analysis & Approaches (SL & HL), part of Functions. Each question has the answer and the marking points an examiner would look for.

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Functions, composites & inverses practice questions

Question 1Structured · 9 marksA* challenge

Let f(x) = x² − 6x, for x ≥ k.

(a)

Find the least value of k for which f has an inverse function.

[2]

(b)

For this value of k, find f⁻¹(x), stating its domain.

[4]

(c)

Solve f(x) = f⁻¹(x).

[3]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	k = 3 f is one-to-one on x ≥ vertex (M1) k = 3 A1	2
(b)	f⁻¹(x) = 3 + √x + 9, x ≥ −9 f(x) = (x − 3)² − 9 (A1) interchanging x and y and rearranging M1 f⁻¹(x) = 3 + √x + 9 A1 domain x ≥ −9 A1	4
(c)	x = 7 the graphs meet on y = x, so f(x) = x M1 x² − 7x = 0 ⇒ x = 0 or x = 7 A1 x = 0 < 3 is not in the domain, so x = 7 R1	3

Question 2Structured · 6 marks

Let f(x) = 5 − x² and g(x) = 2^x, for −4 ≤ x ≤ 4.

(a)

Find the x-coordinates of the points where the graphs of f and g intersect.

[3]

(b)

Hence solve the inequality f(x) > g(x).

[3]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	x = −2.19, x = 1.48 attempt to solve f(x) = g(x), e.g. graph or solver (M1) x = −2.19, x = 1.48 A2 award A1 if one value is missing or incorrect	3
(b)	−2.19 < x < 1.48 using the graph to choose the correct regions (M1) correct critical values used as endpoints (A1) −2.19 < x < 1.48 A1	3

Question 3Structured · 6 marks

Let f(x) = −2x + 3 and g(x) = x² − 6, for x ∈ ℝ.

(a)

Find (f∘g)(x).

[2]

(b)

Solve (g∘f)(x) = 30.

[4]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	−2x² + 15 f(x² − 6) = −2(x² − 6) + 3 (M1) −2x² + 15 A1	2
(b)	x = −32, x = 92 (−2x + 3)² − 6 = 30 M1 (−2x + 3)² = 36 (A1) −2x + 3 = ±6 (M1) x = −32, x = 92 A1	4

Question 4Structured · 7 marks

Let f(x) = 4x + 1x − 4, x ≠ 4.

(a)

Write down the equation of the vertical asymptote and the equation of the horizontal asymptote of the graph of f.

[2]

(b)

Find f⁻¹(x).

[3]

(c)

Hence state, with a reason, a special property of f.

[2]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	x = 4, y = 4 x = 4 A1 y = 4 A1	2
(b)	f⁻¹(x) = 4x + 1x − 4 interchanging x and y (M1) x(y − 4) = 4y + 1 ⇒ y(x − 4) = 4x + 1 (A1) f⁻¹(x) = 4x + 1x − 4 A1	3
(c)	f is self-inverse f⁻¹(x) = f(x) R1 f is self-inverse (its graph is symmetric in y = x) A1	2

Question 5Structured · 9 marksA* challenge

Let f(x) = x² − 8x + 14, for x ≥ k.

(a)

Find the least value of k for which f has an inverse function.

[2]

(b)

For this value of k, find f⁻¹(x), stating its domain.

[4]

(c)

Solve f(x) = f⁻¹(x).

[3]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	k = 4 f is one-to-one on x ≥ vertex (M1) k = 4 A1	2
(b)	f⁻¹(x) = 4 + √x + 2, x ≥ −2 f(x) = (x − 4)² − 2 (A1) interchanging x and y and rearranging M1 f⁻¹(x) = 4 + √x + 2 A1 domain x ≥ −2 A1	4
(c)	x = 7 the graphs meet on y = x, so f(x) = x M1 x² − 9x + 14 = 0 ⇒ x = 2 or x = 7 A1 x = 2 < 4 is not in the domain, so x = 7 R1	3

Question 6Structured · 6 marks

Let f(x) = x³ − 4x + 1 and g(x) = e^−x, for −3 ≤ x ≤ 3.

(a)

Find the x-coordinates of the points where the graphs of f and g intersect.

[3]

(b)

Hence solve the inequality f(x) < g(x).

[3]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	x = −1.37, x = 0, x = 1.88 attempt to solve f(x) = g(x), e.g. graph or solver (M1) x = −1.37, x = 0, x = 1.88 A2 award A1 if one value is missing or incorrect	3
(b)	−3 ≤ x < −1.37 or 0 < x < 1.88 using the graph to choose the correct regions (M1) correct critical values used as endpoints (A1) −3 ≤ x < −1.37 or 0 < x < 1.88 A1	3

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in

Functions, composites & inverses: IB Mathematics: Analysis & Approaches questions with answers

Functions, composites & inverses practice questions

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