The function f is defined by f ^xh =
1
1 for x 2 2.5.
2x - 5
(i) Find an expression for f -1 ^xh. [2]
(ii) State the domain of f -1 ^xh. [1]
(iii) Find an expression for f 2 ^xh, giving your answer in the form
ax + b
, where a, b, c and d are
cx + d
integers to be found. [3]
2 The function f is defined by f (x) = ln(2x + 1) for x H 0.
(a) Sketch the graph of y = f (x) and hence sketch the graph of y = f -1 (x) on the axes below. [3]
O x
The function g is defined by g (x) = (x - 4) 2 + 1 for x G 4.
(b) (i) Find an expression for g -1 (x) and state its domain and range. [4]
(ii) Find and simplify an expression for fg (x) . [2]
(iii) Explain why the function gf does not exist. [1]
3 (a) A function f is defined, for all real x, by
f (x) = x - x 2 .
Find the greatest value of f (x) and the value of x for which this occurs. [3]
(b) The domain of g (x) = x - x 2 is such that g -1 (x) exists. Explain why x H 1 is a suitable domain
for g(x). [1]
(c) The functions h and k are defined by
h: x 7 lg (x + 2) for x 2- 2 ,
k: x 7 5 + x - 1 for 1 1 x 1 101.
(i) Find hk(10). [2]
(ii) Find k -1 (x) , stating its domain and range. [5]
3
4 A function f is defined, for x G , by f (x) = 2x 2 - 6x + 5.
2
(i) Express f (x) in the form a (x - b) 2 + c , where a, b and c are constants. [3]
(ii) On the same axes, sketch the graphs of y = f (x) and y = f -1 (x) , showing the geometrical
relationship between them. [3]
O x
(iii) Using your answer from part (i), find an expression for f -1 (x) , stating its domain. [3]
1 3
5 The function g is defined, for x 2- , by g (x) = .
2 2x + 1
(i) Show that gl (x) is always negative. [2]
(ii) Write down the range of g. [1]
The function h is defined, for all real x, by h (x) = kx + 3, where k is a constant.
(iii) Find an expression for hg (x) . [1]
(iv) Given that hg (0) = 5, find the value of k. [2]
(v) State the domain of hg. [1]
6 (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 down a suitable domain for the function f (x) = (x + 3) (x - 5) such that f has an
inverse. [1]
(b) The functions g and h are defined by
g (x) = 3x - 1 for x > 1,
4
h (x) = for x ≠ 0.
x
(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]
7 (a) The functions f and g are defined by
f (x) = 5x - 2 for x 2 1,
g (x) = 4x 2 - 9 for x 2 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 G- 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]
8 The functions f and g are defined for real values of x H 1 by
f (x) = 4x - 3,
2x + 1
g (x) = .
3x - 1
(i) Find gf (x) . [2]
(ii) Find g -1 (x) . [3]
(iii) Solve fg (x) = x - 1. [4]
9
y y
A B
O x O x
C y D y
O x O x
The four graphs above are labelled A, B, C and D.
(i) Write down the letter of each graph that represents a function, giving a reason for your choice. [2]
(ii) Write down the letter of each graph that represents a function which has an inverse, giving a
reason for your choice. [2]
10 The functions f and g are defined, for x 2 1 , by
f (x) = 9 x - 1 ,
g (x) = x 2 + 2 .
(i) Find an expression for f -1 (x) , stating its domain. [3]
(ii) Find the exact value of fg(7). [2]
(iii) Solve gf (x) = 5x 2 + 83x - 95. [4]
11 The function f is defined by f ^xh =
1
for x 2 2.5.
2x - 5
(i) Find an expression for f -1 ^xh. [2]
(ii) State the domain of f -1 ^xh. [1]
(iii) Find an expression for f 2 ^xh, giving your answer in the form
ax + b
, where a, b, c and d are
cx + d
integers to be found. [3]
12 The functions f and g are defined for x 2 1 by
f (x) = 2 + ln x ,
g (x) = 2 e x + 3 .
(i) Find fg (x) . [1]
(ii) Find ff (x) . [1]
(iii) Find g -1 (x) . [2]
(iv) Solve the equation f (x) = 4 . [1]
(v) Solve the equation gf (x) = 20 . [4]
13 The functions f and g are defined for real values of x by
f ^xh = ^x + 2h2 + 1 ,
g ^xh =
x-2 1
, x! .
2x - 1 2
(i) Find f 2 ^- 3h. [2]
(ii) Show that g -1 ^xh = g ^xh. [3]
(iii) Solve gf ^xh =
8
. [4]
19
14 The functions f and g are defined by
2
f (x) = ln (3x + 2) for x 2 - ,
3
2x
g (x) = e - 4 for x ! R .
(i) Solve gf (x) = 5 . [5]
(ii) Find f -1 (x) . [2]
(iii) Solve f -1 (x) = g (x) . [4]
