Question Simulator

IGCSE Additional Mathematics (0606)

Functions Banker

11 questions · about 6% of the marks we indexed · examined in 8 of 9 sessions.

Questions by year

2026

  • 2026 May/June · Paper 22 · Q8 (a, b(i), b(ii), b(iii), c(i), c(ii), c(iii))12 marksComposite functions · Domain and range · Graphs of a function and its inverse · Inverse functions

2025

  • 2025 Oct/Nov · Paper 12 · Q4 (c(i), c(ii))6 marksDomain and range · Graphs of a function and its inverse
  • 2025 Oct/Nov · Paper 22 · Q9 (a, b, c)5 marksComposite functions · Domain and range
  • 2025 May/June · Paper 12 · Q9 (a, b, c, d)8 marksDomain and range · Graphs of a function and its inverse · Inverse functions
  • 2025 May/June · Paper 22 · Q34 marksComposite functions

2024

  • 2024 Oct/Nov · Paper 12 · Q4 (a(i), a(ii), a(iii), a(iv), b)13 marksComposite functions · Domain and range · Graphs of a function and its inverse · Inverse functions
  • 2024 May/June · Paper 22 · Q10 (a, b, c)8 marksComposite functions · Inverse functions

2023

  • 2023 Oct/Nov · Paper 12 · Q8 (a(i), a(ii), a(iii), b)9 marksComposite functions · Domain and range · Graphs of a function and its inverse
  • 2023 Oct/Nov · Paper 12 · Q12 (a)1 markDomain and range
  • 2023 May/June · Paper 12 · Q8 (a, b, c, d(i), d(ii))10 marksComposite functions · Domain and range · Graphs of a function and its inverse

2022

  • 2022 Oct/Nov · Paper 12 · Q8 (a, b, c, d)10 marksDomain and range · Graphs of a function and its inverse · Inverse functions

References only, not question text. Open Official paper ↗ where Cambridge publishes the paper, or see the official Cambridge past-papers page; for other papers, ask your school. “Paper 42” means Paper 4, variant 2.

Practice questions

Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.

Q1Graphs of a function and its inverse · Core · Paper 1 style · no calculator · 5 marks
  1. (a) The function f is defined by f(x) = 2x + 5 for x ∈ ℝ. The graph of y = f(x) passes through (1, 7) and (−2, 1). Write down the coordinates of two points on the graph of y = f⁻¹(x). [2]
  2. (b) Find f⁻¹(x). [2]
  3. (c) Describe the geometrical relationship between the graphs of y = f(x) and y = f⁻¹(x). [1]
Show answer
  1. (a) (7, 1) and (1, −2)
    B1 B1 (swap coordinates)
  2. (b) f⁻¹(x) = (x − 5)/2
    M1 x = 2y + 5; A1
  3. (c) Each is the reflection of the other in the line y = x
    B1
Q2Graphs of a function and its inverse · Core · Paper 1 style · no calculator · 5 marks
  1. (a) f(x) = eˣ + 2 for x ∈ ℝ. State the range of f. [1]
  2. (b) Find f⁻¹(x) and state its domain. [2]
  3. (c) The graph of y = f(x) crosses the y-axis at P. Write down the coordinates of the point where the graph of y = f⁻¹(x) crosses the x-axis, and the equation of the asymptote of y = f⁻¹(x). [2]
Show answer
  1. (a) f(x) > 2
    B1
  2. (b) f⁻¹(x) = ln(x − 2), x > 2
    M1 x − 2 = e^y; A1 with domain
  3. (c) (3, 0); x = 2
    B1 P = (0, 3) reflects to (3, 0); B1 x = 2
Q3Graphs of a function and its inverse · Core · Paper 2 style · calculator · 6 marks
  1. (a) g(x) = ln(2x − 1) for x > ½. Find the coordinates of the point where y = g(x) crosses the x-axis, and state the equation of its asymptote. [2]
  2. (b) Find g⁻¹(x). [2]
  3. (c) Write down the coordinates of the point where y = g⁻¹(x) crosses the y-axis and the equation of its asymptote. [2]
Show answer
  1. (a) (1, 0); x = ½
    B1 B1
  2. (b) g⁻¹(x) = (eˣ + 1)/2
    M1 2y − 1 = eˣ; A1
  3. (c) (0, 1); y = ½
    B1 B1
Q4Graphs of a function and its inverse · Core · Paper 2 style · calculator · 5 marks
  1. (a) f(x) = √(x + 4) for x ≥ −4. State the range of f. [1]
  2. (b) Find f⁻¹(x) and state its domain. [2]
  3. (c) The graph of y = f(x) passes through (−4, 0), (0, 2) and (5, 3). Write down three points on the graph of y = f⁻¹(x), and find f⁻¹(3). [2]
Show answer
  1. (a) f(x) ≥ 0
    B1
  2. (b) f⁻¹(x) = x² − 4, x ≥ 0
    M1; A1
  3. (c) (0, −4), (2, 0), (3, 5); f⁻¹(3) = 5
    B1; B1
Q5Graphs of a function and its inverse · Standard · Paper 1 style · no calculator · 7 marks
  1. (a) f(x) = (x − 2)² + 1 for x ≥ 2. Find f⁻¹(x) and state its domain. [3]
  2. (b) Find the exact coordinates of the point where the graphs of y = f(x) and y = f⁻¹(x) meet. [4]
Show answer
  1. (a) f⁻¹(x) = 2 + √(x − 1), x ≥ 1
    M1 x − 1 = (y − 2)²; A1 positive root since y ≥ 2; B1 x ≥ 1
  2. (b) ((5 + √5)/2, (5 + √5)/2)
    M1 they meet on y = x: (x − 2)² + 1 = x; M1 x² − 5x + 5 = 0; A1 x = (5 ± √5)/2; A1 reject (5 − √5)/2 < 2
Q6Graphs of a function and its inverse · Standard · Paper 1 style · no calculator · 7 marks
  1. (a) f(x) = 4 − e^(2x) for x ∈ ℝ. State the range of f and find the exact coordinates of the points where y = f(x) meets the axes. [3]
  2. (b) Find f⁻¹(x), stating its domain. [2]
  3. (c) Write down the coordinates of the points where y = f⁻¹(x) meets the axes and the equation of its asymptote. [2]
Show answer
  1. (a) f(x) < 4; (0, 3) and (ln 2, 0)
    B1; B1 (0, 3); B1 e^(2x) = 4 ⇒ x = ½ ln 4 = ln 2
  2. (b) f⁻¹(x) = ½ ln(4 − x), x < 4
    M1; A1
  3. (c) (3, 0) and (0, ln 2); x = 4
    B1 B1
Q7Graphs of a function and its inverse · Standard · Paper 2 style · calculator · 7 marks
  1. (a) h(x) = 2 ln(x + 3) for x > −3. Find the coordinates of the points where y = h(x) meets the axes, giving the y-intercept correct to 3 significant figures. [3]
  2. (b) Find h⁻¹(x). [2]
  3. (c) State the equations of the asymptotes of y = h(x) and y = h⁻¹(x). [2]
Show answer
  1. (a) (−2, 0) and (0, 2.20)
    B1 (−2, 0); M1 2 ln 3; A1 2.20
  2. (b) h⁻¹(x) = e^(x/2) − 3
    M1; A1
  3. (c) x = −3 and y = −3
    B1 B1
Q8Graphs of a function and its inverse · Standard · Paper 1 style · no calculator · 5 marks
  1. (a) f(x) = (2x + 1)/(x − 3) for x ≠ 3. Find f⁻¹(x), stating the value of x for which it is not defined. [3]
  2. (b) Write down the equations of the asymptotes of y = f(x) and of y = f⁻¹(x). [2]
Show answer
  1. (a) f⁻¹(x) = (3x + 1)/(x − 2), x ≠ 2
    M1 xy − 3x = 2y + 1; M1 rearrange; A1
  2. (b) f: x = 3, y = 2; f⁻¹: x = 2, y = 3
    B1 B1
Q9Graphs of a function and its inverse · Challenge · Paper 1 style · no calculator · 7 marks
  1. (a) f(x) = x² + 4x for x ≥ −2. Explain why f has an inverse, and find f⁻¹(x), stating its domain. [4]
  2. (b) Find f⁻¹(12). [1]
  3. (c) Find the coordinates of every point where the graphs of y = f(x) and y = f⁻¹(x) meet. [2]
Show answer
  1. (a) f is one-one on x ≥ −2 (the vertex is at x = −2); f⁻¹(x) = −2 + √(x + 4), x ≥ −4
    B1 one-one reason; M1 (x + 2)² − 4; A1 positive root; B1 x ≥ −4
  2. (b) 2
    B1
  3. (c) (0, 0) only
    M1 x² + 4x = x ⇒ x = 0 or −3; A1 −3 rejected (not ≥ −2)
Q10Graphs of a function and its inverse · Challenge · Paper 2 style · calculator · 6 marks
  1. (a) f(x) = 1 + ln x for x > 0. Find f⁻¹(x). [2]
  2. (b) Show that the graphs of y = f(x) and y = f⁻¹(x) meet at (1, 1). [1]
  3. (c) Find the gradients of both graphs at (1, 1) and hence describe how the two graphs meet there. [3]
Show answer
  1. (a) f⁻¹(x) = e^(x − 1)
    M1; A1
  2. (b) f(1) = 1 + 0 = 1 and f⁻¹(1) = e⁰ = 1
    B1
  3. (c) Both gradients are 1, so the graphs touch at (1, 1) with common tangent y = x
    M1 f′(x) = 1/x = 1; M1 (f⁻¹)′(x) = e^(x − 1) = 1; A1 touch, tangent y = x
Q11Composite functions · Core · Paper 1 style · no calculator · 5 marks
  1. (a) f(x) = 3x − 1 and g(x) = x² + 2 for x ∈ ℝ. Find fg(2). [1]
  2. (b) Find gf(x) in terms of x. [1]
  3. (c) Solve fg(x) = 23. [3]
Show answer
  1. (a) 17
    B1 f(6)
  2. (b) (3x − 1)² + 2
    B1
  3. (c) x = ±√6
    M1 3(x² + 2) − 1 = 23; M1 x² = 6; A1
Q12Composite functions · Core · Paper 1 style · no calculator · 4 marks
  1. (a) f(x) = eˣ and g(x) = 2x + 3 for x ∈ ℝ. Write down fg(x) and gf(x). [2]
  2. (b) Solve gf(x) = 11, giving your answer in exact form. [2]
Show answer
  1. (a) fg(x) = e^(2x + 3); gf(x) = 2eˣ + 3
    B1 B1
  2. (b) x = ln 4 (= 2 ln 2)
    M1 eˣ = 4; A1
Q13Composite functions · Core · Paper 2 style · calculator · 5 marks
  1. (a) f(x) = ln x for x > 0 and g(x) = x² + 1 for x ∈ ℝ. Find fg(3), correct to 3 significant figures. [1]
  2. (b) Find gf(e²). [1]
  3. (c) Solve fg(x) = 3, giving your answers correct to 3 significant figures. [3]
Show answer
  1. (a) 2.30
    B1 ln 10
  2. (b) 5
    B1 2² + 1
  3. (c) x = ±4.37
    M1 x² + 1 = e³; M1 x² = e³ − 1; A1
Q14Composite functions · Standard · Paper 1 style · no calculator · 6 marks
  1. (a) f(x) = 1/(x − 2) for x ≠ 2. Show that ff(x) = (x − 2)/(5 − 2x) and state the value of x, other than 2, for which ff is not defined. [3]
  2. (b) Solve ff(x) = x, giving your answers in exact form. [3]
Show answer
  1. (a) ff(x) = (x − 2)/(5 − 2x); x = 5/2
    M1 1/(1/(x − 2) − 2); A1 simplified; B1 x = 5/2
  2. (b) x = 1 ± √2
    M1 x − 2 = 5x − 2x²; M1 x² − 2x − 1 = 0; A1
Q15Composite functions · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) f(x) = 2x + k and g(x) = x² − 3 for x ∈ ℝ, where k is a constant. Given that gf(1) = 6, find the possible values of k. [3]
  2. (b) For the positive value of k, find fg(x). [1]
Show answer
  1. (a) k = 1 or k = −5
    M1 (2 + k)² − 3 = 6; M1 2 + k = ±3; A1
  2. (b) fg(x) = 2x² − 5
    B1 2(x² − 3) + 1
Q16Composite functions · Standard · Paper 2 style · calculator · 5 marks
  1. (a) f(x) = 4 − x² for x ∈ ℝ and g(x) = 2 ln x for x > 0. Find the set of values of x for which gf(x) is defined. [2]
  2. (b) Find gf(1), correct to 3 significant figures. [1]
  3. (c) Solve gf(x) = 0, giving your answers in exact form. [2]
Show answer
  1. (a) −2 < x < 2
    M1 4 − x² > 0; A1
  2. (b) 2.20
    B1 2 ln 3
  3. (c) x = ±√3
    M1 4 − x² = 1; A1
Q17Composite functions · Challenge · Paper 1 style · no calculator · 6 marks
  1. (a) f(x) = x² + 2x for x ∈ ℝ and g(x) = eˣ − 1 for x ∈ ℝ. Show that fg(x) = e^(2x) − 1. [3]
  2. (b) State the range of fg. [1]
  3. (c) Solve fg(x) = 8, giving your answer in exact form. [2]
Show answer
  1. (a) (eˣ − 1)² + 2(eˣ − 1) = e^(2x) − 2eˣ + 1 + 2eˣ − 2 = e^(2x) − 1
    M1 substitute; M1 expand; A1
  2. (b) fg(x) > −1
    B1
  3. (c) x = ln 3
    M1 e^(2x) = 9; A1
Q18Composite functions · Challenge · Paper 2 style · calculator · 7 marks
  1. (a) f(x) = 3x + 2 for x ∈ ℝ and g(x) = (x + 4)/(x − 1) for x ≠ 1. Show that gg(x) = x and state what this tells you about g⁻¹. [3]
  2. (b) Find gf(x) and solve gf(x) = f(x), giving your answers correct to 3 significant figures. [4]
Show answer
  1. (a) gg(x) = [(x + 4) + 4(x − 1)]/[(x + 4) − (x − 1)] = 5x/5 = x; so g⁻¹ = g (g is self-inverse)
    M1 substitute; A1 simplify; B1
  2. (b) gf(x) = (3x + 6)/(3x + 1); x = 0.412 or x = −1.08
    B1 gf(x); M1 3x + 6 = (3x + 2)(3x + 1); M1 9x² + 6x − 4 = 0; A1 x = (−1 ± √5)/3
Q19Inverse functions · Core · Paper 1 style · no calculator · 3 marks
  1. (a) f(x) = (5x − 2)/3 for x ∈ ℝ. Find f⁻¹(x). [2]
  2. (b) Find f⁻¹(6). [1]
Show answer
  1. (a) f⁻¹(x) = (3x + 2)/5
    M1; A1
  2. (b) 4
    B1
Q20Inverse functions · Core · Paper 1 style · no calculator · 4 marks
  1. (a) f(x) = 3 + 2 ln(x − 1) for x > 1. Find f⁻¹(x) and state its domain. [3]
  2. (b) Find f⁻¹(3). [1]
Show answer
  1. (a) f⁻¹(x) = 1 + e^((x − 3)/2), x ∈ ℝ
    M1 ln(y − 1) = (x − 3)/2; A1; B1 domain = range of f = ℝ
  2. (b) 2
    B1 1 + e⁰
Q21Inverse functions · Core · Paper 2 style · calculator · 4 marks
  1. (a) The function f is defined by f(x) = 7 − 2e^(3x) for x ∈ ℝ. State the range of f. [1]
  2. (b) Find f⁻¹(x), stating its domain. [2]
  3. (c) Find f⁻¹(1), correct to 3 significant figures. [1]
Show answer
  1. (a) f(x) < 7
    B1
  2. (b) f⁻¹(x) = ⅓ ln((7 − x)/2), x < 7
    M1; A1
  3. (c) 0.366
    B1 ⅓ ln 3
Q22Inverse functions · Standard · Paper 1 style · no calculator · 5 marks
  1. (a) f(x) = (3x − 1)/(x + 2) for x ≠ −2. Find f⁻¹(x), stating the value of x for which it is not defined. [3]
  2. (b) Solve f⁻¹(x) = 4. [2]
Show answer
  1. (a) f⁻¹(x) = (2x + 1)/(3 − x), x ≠ 3
    M1 xy + 2x = 3y − 1; M1 collect y; A1
  2. (b) x = 11/6
    M1 2x + 1 = 12 − 4x; A1 (check: f(4) = 11/6)
Q23Inverse functions · Standard · Paper 2 style · calculator · 6 marks
  1. (a) f(x) = 2x² − 8x + 3 for x ≥ 2. Write f(x) in the form a(x + b)² + c. [2]
  2. (b) Find f⁻¹(x) and state its domain. [3]
  3. (c) Find f⁻¹(13). [1]
Show answer
  1. (a) 2(x − 2)² − 5
    B1 a, b; B1 c
  2. (b) f⁻¹(x) = 2 + √((x + 5)/2), x ≥ −5
    M1; A1 positive root; B1
  3. (c) 5
    B1
Q24Inverse functions · Challenge · Paper 1 style · no calculator · 4 marks
  1. (a) f(x) = ln(x² − 4) for x > 2. Find f⁻¹(x) and state its domain. [3]
  2. (b) Find the exact value of f⁻¹(ln 5). [1]
Show answer
  1. (a) f⁻¹(x) = √(eˣ + 4), x ∈ ℝ
    M1 x² − 4 = e^y; A1 positive root; B1
  2. (b) 3
    B1 √9
Q25Inverse functions · Challenge · Paper 2 style · calculator · 6 marks
  1. (a) f(x) = e^(2x) − 4eˣ for x ≥ ln 2. Show that f(x) = (eˣ − 2)² − 4 and state the range of f. [2]
  2. (b) Find f⁻¹(x) and state its domain. [3]
  3. (c) Find the exact value of f⁻¹(5). [1]
Show answer
  1. (a) f(x) ≥ −4
    B1 expand; B1
  2. (b) f⁻¹(x) = ln(2 + √(x + 4)), x ≥ −4
    M1 eʸ − 2 = √(x + 4); A1; B1
  3. (c) ln 5
    B1
Q26Domain and range · Core · Paper 1 style · no calculator · 4 marks
  1. (a) f(x) = x² − 4x + 7 for x ∈ ℝ. By completing the square, find the range of f. [2]
  2. (b) g(x) = 3 − 2x for 0 ≤ x ≤ 4. Find the range of g. [2]
Show answer
  1. (a) f(x) ≥ 3
    M1 (x − 2)² + 3; A1
  2. (b) −5 ≤ g(x) ≤ 3
    B1 g(0) = 3; B1 g(4) = −5
Q27Domain and range · Core · Paper 2 style · calculator · 3 marks
  1. (a) The amount of a medicine in a patient's blood is modelled by f(t) = 5e^(−t) + 1 for t ≥ 0. Find the range of f. [2]
  2. (b) g(x) = ln(3x − 6). State the largest possible domain of g. [1]
Show answer
  1. (a) 1 < f(t) ≤ 6
    B1 f(0) = 6; B1 f(t) > 1 as e^(−t) > 0
  2. (b) x > 2
    B1
Q28Domain and range · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) f(x) = 3 cos 2x − 1 for 0 ≤ x ≤ π. Find the range of f. [2]
  2. (b) h(x) = √(9 − x²). State the largest possible domain of h and the corresponding range. [2]
Show answer
  1. (a) −4 ≤ f(x) ≤ 2
    B1 max 3 − 1; B1 min −3 − 1
  2. (b) Domain −3 ≤ x ≤ 3; range 0 ≤ h(x) ≤ 3
    B1 B1
Q29Domain and range · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) f(x) = (x + 1)² − 9 for −4 ≤ x ≤ 2. Find the range of f. [2]
  2. (b) Explain why f does not have an inverse. [1]
  3. (c) The domain is changed to x ≥ k so that f has an inverse. State the least possible value of k. [1]
Show answer
  1. (a) −9 ≤ f(x) ≤ 0
    B1 min −9 at x = −1; B1 max 0 at x = −4 and x = 2
  2. (b) f is not one-one, e.g. f(−4) = f(2) = 0
    B1
  3. (c) k = −1
    B1
Q30Domain and range · Challenge · Paper 2 style · calculator · 6 marks
  1. (a) f(x) = √(x − 1) for x ≥ 1 and g(x) = 5 − x² for x ∈ ℝ. Find the set of values of x for which fg(x) is defined. [2]
  2. (b) Find the range of fg for these values of x. [2]
  3. (c) Find gf(x), simplifying your answer, and state the range of gf. [2]
Show answer
  1. (a) −2 ≤ x ≤ 2
    M1 5 − x² ≥ 1; A1
  2. (b) 0 ≤ fg(x) ≤ 2
    B1 max √4 at x = 0; B1 min 0 at x = ±2
  3. (c) gf(x) = 6 − x; gf(x) ≤ 5
    B1 5 − (x − 1); B1 x ≥ 1 ⇒ ≤ 5

Similar questions

Questions that test the same subtopic in different years, grouped so you can work on one skill at a time.

References only: find each question in its paper (official link where available, otherwise through your school).

Graphs of a function and its inverse 7 questions · 7/9 sessions
Composite functions 7 questions · 7/9 sessions
Inverse functions 5 questions · 5/9 sessions
Domain and range 9 questions · 7/9 sessions

Subtopics

Our analysis of how often each subtopic was examined and how many marks it carried, 2022 to May/June 2026.

SubtopicMarks indexedSessions
Graphs of a function and its inverse 297/9
Composite functions 237/9
Inverse functions 195/9
Domain and range 157/9

← All Additional Mathematics topics