1
1 Find the set of values of k for which the line y = k ^4 x - 3h does not intersect the curve
y = 4x 2 + 8x - 8. [5]
2
2 (i) Sketch the graph of y = ^2x + 1h^x - 2h for - 2 G x G 3, showing the coordinates of the
points where the curve meets the x- and y-axes. [3]
(ii) Find the non-zero values of k for which the equation ^2x + 1h^x - 2h = k has two solutions only.
[2]
3
3 (i) On the axes below, sketch the graph of y = ^x - 4h^x + 2h showing the coordinates of the points
where the curve meets the x-axis. [2]
O x
(ii) Find the set of values of k for which ^x - 4h^x + 2h = k has four solutions. [3]
4
4 (i) Express 2x 2 - x + 6 in the form p ^x - qh2 + r , where p, q and r are constants to be found. [3]
(ii) Hence state the least value of 2x 2 - x + 6 and the value of x at which this occurs. [2]
5
5 The functions f and g are defined by
2x
f ^xh = for x 2 0 ,
x+1
g ^xh = x + 1 for x 2-1.
(i) Find fg ^8h. [2]
ax
(ii) Find an expression for f 2 ^xh, giving your answer in the form , where a, b and c are integers
bx + c
to be found. [3]
(iii) Find an expression for g -1 ^xh, stating its domain and range. [4]
6
(iv) On the same axes, sketch the graphs of y = g ^xh and y = g -1 ^xh, indicating the geometrical
relationship between the graphs. [3]
O x
7
6 (i) Express 12x 2 - 6x + 5 in the form p ^x - qh2 + r , where p, q and r are constants to be found.
[3]
1
(ii) Hence find the greatest value of and state the value of x at which this
12x - 6x + 5
2
occurs. [2]
8
7 The functions f and g are defined, for real values of x greater than 2, by
f (x) = 2 x - 1,
g (x) = x ^x + 1h.
(i) State the range of f. [1]
(ii) Find an expression for f -1 (x) , stating its domain and range. [4]
9
(iii) Find an expression for gf (x) and explain why the equation gf (x) = 0 has no solutions. [4]
10
8 It is given that f ^xh = 3e 2x for x H 0 ,
g ^xh = ^x + 2h2 + 5 for x H 0 .
(i) Write down the range of f and of g. [2]
(ii) Find g -1 , stating its domain. [3]
(iii) Find the exact solution of gf ^xh = 41. [4]
11
(iv) Evaluate f l^ln 4h. [2]
12
9 (a)
y
O 2 x
The diagram shows the graph of y = f (x) passing through ^0, 4h and touching the x-axis at ^2, 0h.
Given that the graph of y = f (x) is a straight line, write down the two possible expressions for f (x) .
[2]
(b) On the axes below, sketch the graph of y = e -x + 3, stating the coordinates of any point of
intersection with the coordinate axes. [3]
O x
13
10 (a) Find the set of values of x for which 4x 2 + 19x - 5 G 0 . [3]
(b) (i) Express x 2 + 8x - 9 in the form ^x + ah2 + b , where a and b are integers. [2]
(ii) Use your answer to part (i) to find the greatest value of 9 - 8x - x 2 and the value of x at
which this occurs. [2]
14
(iii) Sketch the graph of y = 9 - 8x - x 2 , indicating the coordinates of any points of intersection
with the coordinate axes. [2]
O x
16
The function f is defined by f (x) = 2 - x + 5 for - 5 G x 1 0 .
11
(i) Write down the range of f. [2]
(ii) Find f -1 (x) and state its domain and range. [4]
4
The function g is defined by g (x) = for - 5 G x 1 - 1 .
x
(iii) Solve fg (x) = 0 . [3]
17
12 (i) Express 4x 2 + 8x - 5 in the form p ^x + qh2 + r , where p, q and r are constants to be found. [3]
(ii) State the coordinates of the vertex of y = 4x 2 + 8x - 5 . [2]
(iii) On the axes below, sketch the graph of y = 4x 2 + 8x - 5 , showing the coordinates of the points
where the curve meets the axes. [3]
O x
18
13 (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]
19
14 (a) It is given that f ^xh = 3e - 4x + 5 for x ! R .
(i) State the range of f. [1]
(ii) Find f -1 and state its domain. [4]
(b) It is given that g ^xh = x 2 + 5 and h ^xh = ln x for x 2 0 . Solve hg ^xh = 2 . [3]
20
15 (i) On the axes below sketch the graphs of y = 2x - 5 and 9y = 80x - 16x 2 . [5]
y
10
–1 0 1 2 3 4 5 6 x
–5
–10
(ii) Solve 2x - 5 = 4 . [3]
(iii) Hence show that the graphs of y = 2x - 5 and 9y = 80x - 16x 2 intersect at the points
where y = 4 . [1]
(iv) Hence find the values of x for which 9 2x - 5 G 80x - 16x 2 . [2]
21
3
16 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]
22
1 3
17 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]
23
18 The functions f and g are defined for real values of x by
f (x) = x-1-3 for x 2 1,
x-2
g (x) = for x 2 2.
2x - 3
(i) Find gf(37). [2]
(ii) Find an expression for f - 1 (x) . [2]
(iii) Find an expression for g -1 (x) . [2]
24
19 The functions f and g are defined for real values of x by
2
f ^xh = + 1 for x 2 1,
x
g ^xh = x 2 + 2 .
Find an expression for
(i) f -1 ^xh, [2]
(ii) gf ^xh, [2]
(iii) fg ^xh. [2]
25
3x + 2
(iv) Show that ff ^xh = and solve ff ^xh = x . [4]
x+2
26
20 Find the range of values of k for which the equation kx 2 + k = 8x - 2xk has 2 real distinct roots. [4]
27
21 (a) A function f is such that f ^xh = x 2 + 6x + 4 for x H 0 .
(i) Show that x 2 + 6x + 4 can be written in the form ^x + ah2 + b , where a and b are integers.
[2]
(ii) Write down the range of f. [1]
(iii) Find f - 1 and state its domain. [3]
28
(b) Functions g and h are such that, for x d R ,
g ^xh = e x and h ^xh = 5x + 2 .
Solve h 2 g ^xh = 37 . [4]
29
22 (i) On the axes below, sketch the graph of y = x 2 - 4x - 12 showing the coordinates of the
pointswherethegraphmeetstheaxes. [3]
O x
(ii) Findthecoordinatesofthestationarypointonthecurve y = x 2 - 4x - 12 . [2]
(iii) Findthevaluesofksuchthattheequation x 2 - 4x - 12 = k hasonly2solutions. [2]
30
23 Find the values of k for which the line y = 2x + k + 2 cuts the curve y = 2x 2 + (k + 2) x + 8
in two distinct points. [6]
31
24 Given that f (x) = 3x 2 + 12x + 2 ,
(i) find values of a, b and c such that f (x) = a (x + b) 2 + c , [3]
(ii) state the minimum value of f(x) and the value of x at which it occurs, [2]
1
(iii) solve f c m = 0 , giving each answer for y correct to 2 decimal places. [3]
y
32
25 (i) Given that 3x 2 + p ^1 - 2xh =- 3, show that, for x to be real, p 2 - 3p - 9 H 0 . [3]
(ii) Hence find the set of values of p for which x is real, expressing your answer in exact form. [3]
33
26 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]
34
(iv) Solve the equation f (x) = 4 . [1]
(v) Solve the equation gf (x) = 20 . [4]
35
27 Functions f and g are defined, for x 2 0 , by
f (x) = ln x ,
g (x) = 2x 2 + 3.
(i) Write down the range of f. [1]
(ii) Write down the range of g. [1]
(iii) Find the exact value of f -1 g (4) . [2]
(iv) Find g -1 ^xh and state its domain. [3]
36
28 Find the set of values of k for which the equation kx 2 + 3x - 4 + k = 0 has no real roots. [4]
r
4 The graph of y = a cos (bx) + c has an amplitude of 3, a period of and passes through the point
4
r 5
c , m. Find the value of each of the constants a, b and c. [4]
12 2
37
29 The functions f and g are defined for real values of x by
f ^xh = ^x + 2h2 + 1 ,
x-2 1
g ^xh = , x! .
2x - 1 2
(i) Find f 2 ^- 3h. [2]
(ii) Show that g -1 ^xh = g ^xh. [3]
8
(iii) Solve gf ^xh = . [4]
19
