PAST PAPERS(2018 – 2023)
Paper 2
Aung
0606
CHAPTER 1:
FUNCTIONS
0606 Paper 2
1. (a) The function f is defined by f(x) = √1 + 𝑥 2 , for all real values of x. The graph of
y = f(x) is given below.
y 1 + x2
O x
(i) Explain, with reference to the graph, why f does not have an inverse. [1]
(ii) Find f 2 (x). [2]
(b) The function g is defined, for x > k, by g(x) = √1 + 𝑥 2 and g has an inverse.
(i) Write down a possible value for k. [1]
(ii) Find g −1 (x). [2]
0606/22/F/M/18 Q (10)
2. (a) (i) On the axes below, sketch the graph of y = |(𝑥 + 3)(𝑥 − 5)| showing the
coordinates of the points where the curve meets the x-axis. [2]
O x
(ii) Write the suitable domain for the function y = |(𝑥 + 3)(𝑥 − 5)| such that f has an inverse. [1]
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0606 Paper 2
(b) The functions g and h are defined by
g(x) = 3x – 1 for x > 1,
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h(x) = 𝑥 for x ≠ 0.
(i) Find hg (x). [1]
(ii) Find (hg)−1(x). [2]
(c) Given that p(a) = b and that the function p has an inverse, write down p−1(b). [1]
0606/22/M/J/18 Q (10)
3. The functions f and g are defined for real values of x ≥ 1 by
f(x) = 4x – 3 ,
2𝑥+1
g(x) = .
3𝑥−1
(i) Find gf(x). [2]
(ii) Find g −1 (x). [2]
(iii) Solve fg(x) = x – 1. [4]
0606/22/O/N/18 Q (11)
4. (a) It is given that g(x) = 6x4 + 5 for all real x.
(i) Explain why g is a function but does not have an inverse. [2]
(ii) Find g2 (x) and state its domain. [2]
It is given that h(x) = 6x4 + 5 for x ≤ k.
(iii) State the greatest value of k such that h−1 exits. [1]
(iv) For this value of k, find h−1(x). [3]
0606/22/F/M/19 Q (9 a)
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0606 Paper 2
5. (a) The functions f and g are defined by
f(x) = 5x – 3 for x > 1 ,
g(x) =4x2 – 9 for x > 0.
(i) State the range of g. [1]
(ii) Find the domain of gf. [1]
(iii) Showing all your working, find the exact solutions of gf(x) = 4. [3]
(b) The function h is defined by h(x) = √𝑥 2 − 1 for x ≤ – 1.
(i) State the geometrical relationship between the graphs of y = h(x) and y = h−1 (x). [1]
(ii) Find an expression for h−1 (x). [3]
0606/22/M/J/19 Q (12)
6. (a) On the axes below, sketch the graph of y = |5x – 7| showing the coordinates of the points where the
graph meets the x-axis. [3]
O x
(b) Solve 5|5x – 7| – 1 = 14. [3]
0606/22/F/M/20 Q (5)
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7. (a) g(x) = 3 + for x ≥ 1.
𝑥
(i) Find an expression for g −1 (x). [2]
(ii) Write down the range of g −1 . [1]
(iii) Find the domain of g −1 .
0606/22/F/M/20 Q (10.a.)
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0606 Paper 2
√4𝑥 2 −1
8. The function f is defined by f(x) = for 0.5 ≤ x ≤ 1.5.
2𝑥
The diagram shows a sketch of y = f(x).
y
4x 2 1
y
2x
0 0.5 1.5 x
(a) (i) It is given that f −1 exists. Find the domain and range of f −1 . [3]
(ii) Find an expression for f −1 (x). [3]
0606/22/F/M/21 Q (10.a.)
9. The functions f and g are defined, for of x > 0, by
2𝑥 2 −1
f(x) = ,
3𝑥
1
g(x) = .
𝑥
(a) Find an simplify an expression for fg(x). [2]
(b) (i) Given that f −1 exits, write down the range of f −1 . [1]
𝑝𝑥+ √𝑞𝑥 2 +𝑟
(iii) Show that f −1(x) = , where p, q and r are integers. [4]
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0606/22/M/J/21 Q (13)
