Functions: AS Level Mathematics questions with answers
Free Functions questions for AS Level Mathematics (9709), part of Pure Mathematics 1. Each question has the answer and the marking points an examiner would look for.
The function f is defined by f(x) = 5x + 3x − 5 for x > 5.
(a)
Find f−1(x).
[3]
(b)
Hence explain what your answer to (a) shows about the graph of y = f(x), and find ff(x).
[2]
(c)
State the range of f.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
5x + 3x − 5
y(x − 5) = 5x + 3 rearranged M1 x terms collected and factorised M1 correct inverse A1
3
(b)
f is self-inverse: the graph is symmetrical about y = x; ff(x) = x
f−1 = f, symmetrical in y = x B1 ff(x) = x B1
2
(c)
f(x) > 5
f(x) > 5 B1
1
Question 2Structured · 8 marksA* challenge
The function f is defined by f(x) = 6x + 2 − 3 for x > −2.
(a)
State the range of f.
[1]
(b)
Find an expression for f−1(x) and state its domain.
[4]
(c)
Solve the equation f−1(x) = 1.
[2]
(d)
Explain why the graphs of y = f(x) and y = f−1(x) are reflections of each other in y = x.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
f(x) > −3
f(x) > −3 B1
1
(b)
f−1(x) = 6x + 3 − 2, x > −3
y = … rearranged to make x the subject M1 x + 2 = 6y + 3A1 f−1(x) = 6x + 3 − 2 A1 domain x > −3 B1
4
(c)
x = −1
x = f(1) M1 x = −1 A1
2
(d)
inverse swaps x and y
(x, y) on f ⇔ (y, x) on f−1: swapping x and y is reflection in y = x B1
1
Question 3Structured · 4 marks
Functions f and g are defined by f(x) = 3x + 4 and g(x) = 5x for x ≠ 0.
(a)
Find fg(x) and gf(x).
[2]
(b)
Solve fg(x) = 232.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
fg(x) = 15x + 4; gf(x) = 53x + 4
fg correct B1 gf correct B1
2
(b)
x = 2
15x = 152M1 x = 2 A1
2
Question 4Structured · 2 marks
The graph of y = f(x) is transformed by a stretch parallel to the x-axis with factor ½ followed by a translation of 0−1. Write down the equation of the new graph in terms of f.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
y = f(2x) − 1
one transformation correct B1 y = f(2x) − 1 B1
2
Question 5Structured · 9 marksA* challenge
The functions f and g are defined for x ∈ ℝ by f(x) = 3x − 5 and g(x) = x2 − 3.
(a)
Find expressions for fg(x) and gf(x).
[2]
(b)
Solve the equation fg(x) = gf(x).
[3]
(c)
Find the set of values of x for which gf(x) > fg(x).
gf(x) − fg(x) = 6(x − α)(x − β) > 0 M1 x < 2, x > 3 A1
2
(d)
f−1(x) = x + 53; x = 52
f−1(x) = x + 53B1 x = 52 (where f(x) = x) B1
2
Question 6Structured · 8 marksA* challenge
The function f is defined by f(x) = x2 − 6x + 7. The graph of y = f(x) is transformed to the graph of y = g(x) by a translation of −20, followed by a stretch parallel to the y-axis with factor 3, followed by a translation of 02.
(a)
Find g(x), giving your answer in the form ax2 + bx + c.
[4]
(b)
Find the coordinates of the vertex of the graph of y = g(x).
[2]
(c)
The order of the stretch and the final translation is reversed. Find the new vertex and explain why it differs.
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in