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Functions · CAPE Pure Mathematics Unit 1

121 past-paper questions on Functions, part of Basic Algebra and Functions, from every CAPE Pure Mathematics Unit 1 paper on Quelpr.

Practise Functions
  1. Q121 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1Which of the following mapping diagrams does NOT represent a function?
  2. Q151 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The sketch below shows a function y = f(x). The function y = |f(x)| is represented by
  3. Q81 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1Which of the following sets of ordered pairs represent functions?\nI. \{(-1, 1), (0, 2), (1, 3), (4, 6)\}\nII. \{(-2, 4), (1, 1), (1, 4), (2, 4)\}\nIII. \{(-1, 1), (0, 0), (1, 1), (-3, 9)\}\nIV.…
  4. Q91 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1If g(x) is the inverse of f(x) then the correct diagram is
  5. Q101 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1If f(x) = 3x - 4 and fg(x) = x, then g(x) is
  6. 2(a)(i)3 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Let f(x) = \frac{3x + 1}{x} and g(x) = e^{-2x} + 1. Show that f^{-1}(x) = \frac{1}{x - 3}.
  7. 2(a)(ii)2 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Hence, or otherwise, write an expression for (f^{-1} \circ g)(x).
  8. 1(a)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1From the graph, state the value of EACH of f(0) and f(2).
  9. 5(a)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Show that f is one-to-one (injective).
  10. 5(b)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Find the value(s) of x in R such that f(f(x)) = f(x) + 6.
  11. 4(a)(i)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find the value of p and of q.
  12. 4(a)(ii)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find the range of the function f(x) for the given domain.
  13. 4(b)(i)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Determine whether f(x) is surjective (onto).
  14. 4(b)(ii)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Determine whether f(x) is injective (one-to-one).
  15. 4(b)(iii)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Determine whether f(x) has an inverse.
  16. Q31 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1Which of the following BEST represents f(x) = x(1 - x)?
  17. Q41 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The tables below show the values of the functions f and g. \begin{tabular}{|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 7 & 5 & 3 & 2 & -7 & -5 \\ \hline \end{tabular}…
  18. Q101 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The point (1, 1) lies on the curve y = f(x). Under which of the following transformations is (1, 1) invariant? \begin{align*} \text{I.} & \quad y = f^{-1}(x) \\ \text{II.} & \quad y = f(-x) \\ \text{III.} & \quad…
  19. Q141 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1Two functions, f and g, are defined as f: x \to 2x + 1, 0 < x < 10, and g: x \to x^2, 0 < x < 3. The composite function gf can only be formed if the domain of f is restricted to
  20. Q41 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1Given that f(x) = x^2, which of the following graphs shows y = f(x) - 1?
  21. Q101 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1The coordinates of the point P are (4, -3). Under a one-way stretch by scale factor 2 in the y-direction with the x-axis invariant, the image of P would be
  22. Q31 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1Which of the following diagrams BEST represents f(x) = x(1 - x)?
  23. Q311 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1In the diagram above showing y^2 = x, y is NOT defined for
  24. Q391 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1From the diagram above, which of the following statements are true? \begin{align*} \text{I.} & \quad f(1) < 0 \\ \text{II.} & \quad f(1) > k \\ \text{III.} & \quad f(2) = 0 \\ \text{IV.} & \quad f(2) = k \end{align*}
  25. Q101 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1The coordinates of the point P are (4, -3). Under a one-way stretch by scale factor 2 in the y-direction with the x-axis invariant, the image of P would be
  26. Q381 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1In the graph showing y^2 = x, y is NOT defined for
  27. Q81 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1Which of the following mapping diagrams does NOT represent a function?
  28. Q91 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1If g(x) is the inverse of f(x) then the correct diagram is
  29. Q111 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1Which of the following sets of ordered pairs represent functions? \begin{align*} \text{I.} & \quad \{(-1, 1), (0, 2), (1, 3), (4, 6)\} \\ \text{II.} & \quad \{(-2, 4), (1, 1), (1, 4), (2, 4)\} \\ \text{III.} & \quad…
  30. Q151 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1The tables below show the values for two functions, f and g. \begin{array}{|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 7 & 5 & 3 & 2 & -7 & -5 \\ \hline \end{array}…
  31. Q91 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1If g(x) is the inverse of f(x) then the correct diagram is
  32. Q151 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1If f(x) = 3x - 4 and f(g(x)) = x, then g(x) is
  33. 2(a)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Let f(x) = 7x + 2. Prove that f is bijective.
  34. 2(c)(i)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2On the axes provided, sketch and label the graph of g(x) = |x^2 + 6x + 8|.
  35. 2(c)(ii)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2On the same axes, sketch and label the inverse of f for x \ge -3.
  36. 2(d)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Given that g(x) = (2x + 3)/(x + 3), prove that g^{-1}(2) does not exist.
  37. 1(a)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Determine the domain of the composite function f(g(x)).
  38. 1(d)8 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Prove that the function f(x) = 3x - 2 is bijective.
  39. 5(a)(i)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Determine f(-2).
  40. 2(b)(i)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2On the grid provided on page 10, sketch the graph of y = |f(x)|.
  41. 2(b)(ii)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2On the grid provided on page 10, sketch the graph of y = |g(x)|.
  42. 2(c)(i)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine whether f is injective.
  43. 2(c)(ii)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Assuming that f is a bijective function, determine an expression for f⁻¹(x).
  44. 5(b)(i)a)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine g(1).
  45. 1(b)6 marks· Pure Mathematics · Unit 1 Q1 1(b)Show that f is one-to-one.
  46. 1(c)7 marks· Pure Mathematics · Unit 1 Q1 1(c)Determine whether there is an x ∈ A such that f(x) = 1.
  47. 1(d)(i)3 marks· Pure Mathematics · Unit 1 Q1 1(d)(i)Explain clearly why f is not 1 – 1.
  48. 1(d)(i)5 marks· Pure Mathematics · Unit 1 Q1 1(d)(i)Use Part (c) above to determine the range of f.
  49. 1(d)(ii)5 marks· Pure Mathematics · Unit 1 Q1 1(d)(ii)Calculate and simplify g f(x).
  50. 1(d)(ii)4 marks· Pure Mathematics · Unit 1 Q1 1(d)(ii)Use Part (c) above to determine whether or not f is onto.
  51. 2(a)7 marks· Pure Mathematics · Unit 1 Q2 2(a)Show that f is one to one.
  52. 2(a)1 mark· Pure Mathematics · Unit 1 Q2 2(a)Find
  53. 2(a)(i)2 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)Evaluate f(-2).
  54. 2(a)(i)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)Use this diagram to assist you in sketching the function x = f(x - 1).
  55. 2(a)(i)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)State the range of f.
  56. 2(a)(i)4 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)Show that (g° f) is one-to-one.
  57. 2(a)(i)2 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)On the diagram above, sketch the inverse of f(x).
  58. 2(a)(i)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)in terms of x, f(f(x)).
  59. 2(a)(i)a)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)a)Determine, in terms of x, f²(x)
  60. 2(a)(i)a)4 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)a)Sketch the inverse of f.
  61. 2(a)(i)b)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)b)Determine, in terms of x, f[g(x)].
  62. 2(a)(i)b)1 mark· Pure Mathematics · Unit 1 Q2 2(a)(i)b)Show that the inverse of f is a function.
  63. 2(a)(ii)4 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)Show that (g° f) is onto.
  64. 2(a)(ii)5 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)Hence, sketch the graph of f(x) = 12x - 2x², showing clearly its main features.
  65. 2(a)(ii)4 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)Calculate the exact values of x which map onto themselves under the function f.
  66. 2(a)(ii)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)Sketch the graph of f.
  67. 2(a)(ii)2 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)Sketch the graph of f(x).
  68. 2(a)(ii)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)Use a graphical method to show that f is bijective.
  69. 2(a)(ii)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)Use this diagram to assist you in sketching the function y = f(x) + 3.
  70. 2(a)(ii)6 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)Determine the values of x for which f(f(x)) = f(x + 3).
  71. 2(a)(ii)1 mark· Pure Mathematics · Unit 1 Q2 2(a)(ii)Hence, or otherwise, state the relationship between f and g.
  72. 2(a)(ii)4 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)Prove that f is one to one.
  73. 2(a)(iii)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(iii)Explain why the inverse function f⁻¹ of f exists.
  74. 2(a)(iii)2 marks· Pure Mathematics · Unit 1 Q2 2(a)(iii)Show that f(x) = 12x - 2x² is NOT one-to-one.
  75. 2(a)(iii)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(iii)Use this diagram to assist you in sketching the function y = |f(x)|.
  76. 2(a)(iii)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(iii)Determine whether f is onto.
  77. 2(a)(iv)2 marks· Pure Mathematics · Unit 1 Q2 2(a)(iv)State the domain of f⁻¹.
  78. 2(a)(v)4 marks· Pure Mathematics · Unit 1 Q2 2(a)(v)Sketch f⁻¹ on the same diagram as f.
  79. 2(b)4 marks· Pure Mathematics · Unit 1 Q2 2(b)Express f as a set of ordered pairs.
  80. 2(b)8 marks· Pure Mathematics · Unit 1 Q2 2(b)Determine whether f is bijective, that is, both one-to-one and onto.
  81. 2(b)4 marks· Pure Mathematics · Unit 1 Q2 2(b)State clearly the transformation which maps g onto f.
  82. 2(b)(i)1 mark· Pure Mathematics · Unit 1 Q2 2(b)(i)Find
  83. 2(b)(i)2 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)On the diagram, insert the asymptotes for the function f.
  84. 2(b)(i)3 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)Sketch the graph of f: A → B.
  85. 2(b)(i)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)Find the value of f(3)
  86. 2(b)(i)1 mark· Pure Mathematics · Unit 1 Q2 2(b)(i)Determine the range of f.
  87. 2(b)(i)a)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)a)f⁻¹(x) and g⁻¹(x)
  88. 2(b)(i)b)1 mark· Pure Mathematics · Unit 1 Q2 2(b)(i)b)f [g(x)] (or f∘g(x)).
  89. 2(b)(ii)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)sketch the graph of f⁻¹, the inverse of f showing the asymptotes for f⁻¹.
  90. 2(b)(ii)2 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)Find the value of f(9)
  91. 2(b)(ii)5 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)Show that (f∘g)⁻¹ (x) = g⁻¹ (x) ∘ f⁻¹ (x).
  92. 2(b)(ii)3 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)Find a set C such that C ⊂ A and f: C → B, is one-to-one.
  93. 2(b)(ii)1 mark· Pure Mathematics · Unit 1 Q2 2(b)(ii)Explain why the function f has an inverse.
  94. 2(b)(ii)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)On a separate diagram, sketch the graph of f(x) = | sin x |, 0 ≤ x ≤ 2π.
  95. 2(b)(ii)a)2 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)a)State TWO reasons why f is NOT a function.
  96. 2(b)(ii)b)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)b)Hence, with MINIMUM changes to f, construct a function g : A → B as a set of ordered pairs.
  97. 2(b)(ii)c)2 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)c)Determine how many different functions are possible for g in (ii) b) above.
  98. 2(b)(iii)3 marks· Pure Mathematics · Unit 1 Q2 2(b)(iii)Find the value of f(-3).
  99. 2(b)(iii)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(iii)By considering the solutions of the equation f(x) = 8, show that f is NOT onto.
  100. 2(b)(iii)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(iii)Find an expression for f⁻¹(x).
  101. 2(b)(iii)a)6 marks· Pure Mathematics · Unit 1 Q2 2(b)(iii)a)By using (b) (ii) above, or otherwise, show that f has an inverse.
  102. 2(b)(iii)b)1 mark· Pure Mathematics · Unit 1 Q2 2(b)(iii)b)By using (b) (ii) above, or otherwise, show that g does NOT have an inverse.
  103. 2(b)(iv)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(iv)By solving the equation f(x) = 0, show that f: A → B is NOT one-to-one.
  104. 2(b)(iv)2 marks· Pure Mathematics · Unit 1 Q2 2(b)(iv)Describe the geometrical relationship between the graphs y = f(x) and y = f⁻¹(x).
  105. 2(b)(v)2 marks· Pure Mathematics · Unit 1 Q2 2(b)(v)Find the range of values of y for which the equation f(x) = y possesses a solution.
  106. 2(c)6 marks· Pure Mathematics · Unit 1 Q2 2(c)Determine the value of f(f⁻¹[f(1)]).
  107. 2(c)3 marks· Pure Mathematics · Unit 1 Q2 2(c)Find the value of f[f(20)].
  108. 2(c)(i)3 marks· Pure Mathematics · Unit 1 Q2 2(c)(i)Find an expression for f(h(x)).
  109. 2(c)(ii)2 marks· Pure Mathematics · Unit 1 Q2 2(c)(ii)Find the value of f[f(8)].
  110. 2(c)(ii)3 marks· Pure Mathematics · Unit 1 Q2 2(c)(ii)Write down the range for the function f(h(x)).
  111. 2(c)(iii)2 marks· Pure Mathematics · Unit 1 Q2 2(c)(iii)Find the value of f[f(3)].
  112. 2(d)(ii)2 marks· Pure Mathematics · Unit 1 Q2 2(d)(ii)State clearly the transformation which maps y = cos x onto y = sin x.
  113. 5(b)(i)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)Determine f(2)
  114. 5(b)(i)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)Sketch the graph of f(x) for the domain -1 ≤ x ≤ 2.
  115. 5(c)(ii)5 marks· Pure Mathematics · Unit 1 Q5 5(c)(ii)Hence, sketch the graph of f.
  116. 5(c)b)4 marks· Pure Mathematics · Unit 1 Q5 5(c)b)Determine the vertical and horizontal asymptotes of f.
  117. 6(b)(i)4 marks· Pure Mathematics · Unit 1 Q6 6(b)(i)Sketch the curve y = x^2 + 4.
  118. 6(b)(v)1 mark· Pure Mathematics · Unit 1 Q6 6(b)(v)Hence, sketch the curve C, showing
  119. 6(b)(v)a)1 mark· Pure Mathematics · Unit 1 Q6 6(b)(v)a)the stationary points
  120. 6(b)(v)b)4 marks· Pure Mathematics · Unit 1 Q6 6(b)(v)b)the points P and Q.
  121. 6(c)(i)3 marks· Pure Mathematics · Unit 1 Q6 6(c)(i)Sketch the curve y = x² + 1.

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