function hl

1.

Given functions f : x Working:

3 l x + 1 and g : x x , find the function (f g) .

Answer: ..................................................................(Total 3 marks)

f ( x) =2. If (a) (b)

x , x + 1 for x 1 and g (x) = (f f )(x), find

g (x) (g g)(2).

Working:

Answers: (a) .................................................................. (b) ..................................................................(Total 3 marks)

1

3.

The function f is defined by

x2 x +1 2 f (x) = x + x + 1(a) (ii) (i) Find an expression for f (x), simplifying your answer.

The tangents to the curve of f (x) at points A and B are parallel to the x-axis. Find the coordinates of A and of B.(5)

(b) (ii) (c)

(i)

Sketch the graph of y = f (x).(5)

Find the x-coordinates of the three points of inflexion on the graph of f.

Find the range of (i) (ii) f; the composite function f f.(5) (Total 15 marks)

2

4.

2 The functions f (x) and g (x) are given by f (x) = x 2 and g (x) = x + x. The function (f g)(x) is defined for x , except for the interval ] a, b [.

(a) (b)

Calculate the value of a and of b. Find the range of f g.

Working:


3

5.

3 The function f is defined by f : x x .

Find an expression for g (x) in terms of x in each of the following cases (a) (b) (f g ) (x) = x + 1; (g f ) (x) = x + 1.

Working:


4

6.

x The functions f and g are defined by f : x e , g : x x + 2.

(a) (b)

Calculate f (3) g (3). Show that (f g) (3) = ln 3 2. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... .....................................................................................................................................(Total 6 marks)1

1

1

5

7.

x +1 Let f and g be two functions. Given that (f g) (x) = 2 and g (x) = 2x 1, find f (x 3).W o r k i n g :

A

n

s w

e r :(Total 6 marks)

6

8.

The functions f and g are defined by f (x) = 2x 1,

x , x 1. g (x) = x + 1Find the values of x for which ( f g) (x) (g f ) (x). .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. ..............................................................................................................................................(Total 6 marks)

7

9.

(a) The function f is defined by f (x) = (x + 2) 3. The function g is defined by g (x) = ax + b, where a and b are constants. Find the value of a, a > 0 and the corresponding value of b, such that

2

3 . f (g (x)) = 4x + 6x 42

(8)

(b)

The functions h and k are defined by h (x) = 5x + 2 and k (x) = cx x + 2 respectively. Find the value of c such that h (k (x)) = 0 has equal roots.(5) (Total 13 marks)

2

8

10.

The functions f and g are defined as:x f (x) = e , x 0 2

1 , x 3. x +3 g (x) =(a) (b) (c) Find h (x) where h (x) = g f (x).(2)

State the domain of h Find h1

1

(x).(2)

(x).(4)

.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. ..............................................................................................................................................(Total 8 marks)

9

11.

4 ,x 2 x+2 Let f (x) = and g (x) = x 1.If h = g f, find (a) (b) h (x);(2)

h (x), where h

1

1

is the inverse of h.(4)

.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. ..............................................................................................................................................(Total 6 marks)

10

12.

Find r =1 Working:

ln( 2 )50 r

, giving the answer in the form a ln 2, where a

.

Answer: ..........................................................................(Total 6 marks)

13.

Find the largest set of values of x such that the function f given by f (x) = values. W o r k i n g :

8x 4 x 3 takes real

A

n

s w

e r :(Total 6 marks)

11

14.

1 Let f be the function f (x) = x arccos x + 2 x for 1 x 1 and g the function g (x) = cos 2xfor 1 x 1. (a) On the grid below, sketch the graph of f and of g. y 2 1 1 0 1 2 3 4 1 x

(b)

Write down the solution of the equation f (x) = g (x).

12

(c)

Write down the range of g.

.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. ..............................................................................................................................................(Total 6 marks)

13

15.

3x 4 , x + 2 x 2. The function f is defined as f (x) =(a) (b) Find an expression for f1

(x).1

Write down the domain of f

.

.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. ..............................................................................................................................................(Total 6 marks)

14

16.

2x The function f is defined by f (x) = x + 6 for x b where b2

.

(a) (b)

12 2 x 2 . ( x 2 + 6) 2 Show that f (x) =

Hence find the smallest exact value of b for which the inverse function f your answer.

1

exists. Justify

.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. ..............................................................................................................................................(Total 6 marks)

15

17.

x2 1 2 The function f is defined for x 0 by f (x) = x + 1 .Find an expression for f (x). Working:1

Answer: .........................................................................(Total 6 marks)

18.

The set of all real numbers under addition is a group ( , +), and the set of all positive + real numbers under multiplication is a group ( , ). Let f denote the mapping of ( , +) to ( + x , ) given by f (x) = 3 . (a) (b) Show that f is an isomorphism of ( , +) onto ( Find an expression for f .(1) (Total 7 marks)1 +

+

, ).(6)

16

19.

The function f is defined for x > 2 by f (x) = ln x + ln (x 2) ln (x 4). x Express f (x) in the form ln x + a .

2

(a) (b)

Find an expression for f (x).

1

W

o

r k i n

g

:

A

n

s w

e r s :

( a ) ( b )(Total 6 marks)

17

20.

The diagram below shows the graph of y1 = f (x). The x-axis is a tangent to f (x) at x = m and f (x) crosses the x-axis at x = n. y y1 = f( x)

0

m

n

x

On the same diagram sketch the graph of y2 = f (x k), where 0 < k < n m and indicate the coordinates of the points of intersection of y2 with the x-axis. Working:

(Total 4 marks)

18

21.

For which values of the real number x is | x + k | = | x | + k, where k is a positive real number? Working:

Answer: ..............(Total 4 marks)

22.

Given that x > 0, find the solution of the following system of equations: 8x 3 = 3 y

9 xy y = x + 42

(Total 3 marks)

19

23.

The diagram below shows the graphs of y = x + 3x and y = g (x), where g (x) is a polynomial of degree 3. y y =3 2

3

2

x

3+

x

g ( A

)x

0 A

x

(a)

If g (2) = 0, g (0) = 4, g (2) = 0, and g (0) = (0) show that g (x) = x + 3x 4. The graph of y = x + 3x is reflected in the y-axis, then translated using the vector 1 1 to give the graph of y = h (x).3 2

3

2

(6)

(b)

Write h (x) in the form h (x) = ax + bx + cx + d.(5)

3

2

The graph of y = x + 3x 4 is obtained by applying a composition of two 3 2 transformations to the graph of y = x + 3x . (c) State the two transformations whose composition maps the graph of y = x + 3x onto 3 2 the graph of y = x + 3x 4 and also maps point A onto point A .3 2

3

2

(3) (Total 14 marks)

20

24.

5 2 Let f (x) = ln |x 3x |, 0.5 < x < 2, x a, x b; (a, b are values of x for which f (x) is not defined).

(a)

(i) Sketch the graph of f (x), indicating on your sketch the number of zeros of f (x). Show also the position of any asymptotes.(2)

(ii) (b) (c) (d)

Find all the zeros of f (x), (that is, solve f (x) = 0).(3)

Find the exact values of a and b.(3)

Find f (x), and indicate clearly where f (x) is not defined. Find the exact value of the x-coordinate of the local maximum of f (x), for 0 < x < 1.5. (You may assume that there is no point of inflexion.)

(3)

(3)

(e)

Write down the definite integral that represents the area of the region enclosed by f (x) and the x-axis. (Do not evaluate the integral.)(2) (Total 16 marks)

21

25.

The following graph is that part of the graph of y = f (x) for which f (x) 0. y 4 3 2 1 2 1 02

1

2

x

Sketch, on the axes provided below, the graph of y = f (x) for 2 x 2. y 4 3 2 1 2 1 0 1 2 3 4(Total 3 marks)

1

2

x

22

26.

(a)2

Sketch and label the curves

1 y = x for 2 x 2, and y = 2 ln x for 0 < x 2.(b) (c) Find the x-coordinate of P, the point of intersection of the two curves.

(2)

(2)

If the tangents to the curves at P meet the y-axis at Q and R, calculate the area of the triangle PQR.(6)

(d)

Prove that the two tangents at the points where x = a, a > 0, on each curve are always perpendicular.(4) (Total 14 marks)

23

27.

(a)

a + b sin x Let y = b + a sin x , where 0 < a < b.(b 2 + a 2 ) cos x dy 2 Show that dx = (b + a sin x) .(4)

(i) (ii)

Find the maximum and minimum values of y.(4)

(iii)

a + b sin x Show that the graph of y = b + a sin x , 0 < a < b cannot have a vertical asymptote.(2)

(b)

For the graph of (i) (ii) (iii)

y = 4 + 5 sin x 5 + 4 sin x for 0 x 2 ,

write down the y-intercept; find the x-intercepts m and n, (where m < n) correct to four significant figures; sketch the graph.(5)

(c)

y = 4 + 5 sin x 5 + 4 sin x and the x-axis from x = 0 to x = n is The area enclosed by the graph of denoted by A. Write down, but do not evaluate, an expression for the area A.(2) (Total 17 marks)

24

28.

The diagram shows a sketch of part of the graph of f (x) = x and a sketch of part of the graph of 2 g (x) = x + 6x 13 y

2

y=

f(

x)

x

y =(

g )

x

(a)

Write down the coordinates of the maximum point of y = g (x).

25

The graph of y = g (x) can be obtained from the graph of y = f (x) by first reflecting the graph of y = f (x), then translating the graph of y = f (x). (b) Describe fully each of these transformations, which together map the graph of y = f (x) onto the graph of y = g (x).

Working:

Answers: (a) .................................................................. (b) .................................................................. ..................................................................(Total 3 marks)

26

29.

The diagram shows the graph of the functions y1 and y2. y 2 0 2 y2

x

y1

y1 On the same axes sketch the graph of y 2 . Indicate clearly where the x-intercepts and asymptotes occur.Working:

(Total 3 marks)

30.

Let (a)

f ( x ) = x 3 ( x 2 1) 2 , 1.4 x 1.4 Sketch the graph of f (x). (An exact scale diagram is not required.) On your graph indicate the approximate position of (i) (ii) (iii) each zero; each maximum point; each minimum point.(4)

27

(b) (ii)

(i)

Find f (x), clearly stating its domain.

Find the x-coordinates of the maximum and minimum points of f (x), for 1 < x < 1.(7)

(c)

Find the x-coordinate of the point of inflexion of f (x), where x > 0, giving your answer correct to four decimal places.(2) (Total 13 marks)

31.

The line segment [AB] has length l, gradient m, (0 < m < 1), and passes through the point (0, 1). It meets the x-axis at A and the line y = x at B, as shown in the diagram. y y= B ( 0 A , 1 O ) x x

(a)

Find the coordinates of A and B in terms of m.(4)

l2 =

(b)

Show that

m2 +1 . m 2 (1 m) 2(3)

(c)

, x 2 (1 x) 2 for x 0, x 1, indicating any asymptotes and the Sketch the graph of coordinates of any maximum or minimum points.(4)

y=

x 2 +1

(d)

Find the value of m for which l is a minimum, and find this minimum value of l.(2) (Total 13 marks)

28

32.

Find the set of values of x for which (e 2) (e 3) 2e . Working:

x

x

x


33.

The graph of the function f (x) = 2x 3x + x + 1 is translated to its image, g (x), by the vector 1 1 . Write g (x) in the form g (x) = ax3 + bx2 +cx + d. Working:

3

2


29

34.

x 2 5x 4 2 Find the equations of all the asymptotes of the graph of y = x 5 x + 4 . Working:

Answer: .......................................................................... ..........................................................................(Total 6 marks)

35.

(a)

On the same axes sketch the graphs of the functions, f (x) and g (x), where f (x) = 4 (1 x) , for 2 x 4, g (x) = ln (x + 3) 2, for 3 x 5.(2)2

(b) (ii) (c) (d)

(i)

Write down the equation of any vertical asymptotes.(3)

State the x-intercept and y-intercept of g (x).

Find the values of x for which f (x) = g (x).(2)

Let A be the region where f (x) g (x) and x 0. (i) (ii) (iii) On your graph shade the region A. Write down an integral that represents the area of A. Evaluate this integral.(4)

(e)

In the region A find the maximum vertical distance between f (x) and g (x).(3) (Total 14 marks)

30

36.

The function f is given by f (x) = 2 x e . Write down (a) (b) the maximum value of f (x); the two roots of the equation f (x) = 0.

2

x

Working:


31

37.

The diagram shows the graph of f (x).

(a)

1 , On the same diagram, sketch the graph of f ( x) indicating clearly any asymptotes.y

2

1

2

1

0 1

1

2

x

2

(b)

On the diagram write down the coordinates of the local maximum point, the local 1 . f ( x) minimum point, the x-intercepts and the y-intercept of

Working:

(Total 6 marks)

32

38.

x 3 arcsin arccos 4 5 , for 4 x 4. Let f (x) = sin (a) On the grid below, sketch the graph of f (x). y 2

1

5

4

3

2

1 0

1

2

3

4

5 x

1

(b)

2

On the sketch, clearly indicate the coordinates of the x-intercept, the y-intercept, the minimum point and the endpoints of the curve of f (x).

(c)

1 Solve f (x) = 2 .

Working:

Answer: (c) ..................................................................(Total 6 marks)

33

39.

For 3 x 3, find the coordinates of the points of intersection of the curves y = x sin x and x + 3y = 1. Working:


40.

k Let f (x) = x k , x k, k > 0(a) On the diagram below, sketch the graph of f. Label clearly any points of intersection with the axes, and any asymptotes. y

k

x

34

(b)

1 On the diagram below, sketch the graph of f . Label clearly any points of intersection with the axes.y

k

x

Working:

(Total 6 marks)

35

41.

Consider the equation e = cos 2x, for 0 x 2. (a) (b) How many solutions are there to this equation? Find the solution closest to 2, giving your answer to four decimal places.

x

Working:


36

42.

The following diagram shows the lines x 2y 4 = 0, x + y = 5 and the point P(1, 1). A line is drawn from P to intersect with x 2y 4 = 0 at Q, and with x + y = 5 at R, so that P is the midpoint of [QR]. y 1 8 6 4 2 1 0 8 6 4 2 Find the exact coordinates of Q and of R. Working: 1 0 2 4 6 8 0 P ( 1 2 , 4 1 ) 6 8 1 0x 0


37

43.

Solve the equation

e 2x

1 x + 2 = 2.

W

o

r k i n

g

:

A

n

s w

e r :

(Total 6 marks)

38

44.

2 2 The graph of y = 2x + 4x +7 is translated using the vector 1 . Find the equation of the translated graph, giving your answer in the form y = ax + bx +c.

.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. ..............................................................................................................................................(Total 6 marks)

39

45.

Each of the diagrams below shows the graph of a function f. Sketch on the given axes the graph of (a)

f (x )

; y

x

(b)

1 f ( x) .y

x

(Total 6 marks)

40

46.

x . The graph of y = cos x is transformed into the graph of y = 8 2 cos 6Find a sequence of simple geometric transformations that does this. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. ..............................................................................................................................................(Total 6 marks)

41

47.

The diagrams below show the graph of y = f (x) which passes through the points A, B, C and D. Sketch, indicating clearly the images of A, B, C and D, the graphs of (a) y = f (x 4);

(2)

(b)

y = f ( 3x).

(4) (Total 6 marks)

42

48.

Determine the values of x that satisfy the following inequalities

x +2(a)

x 3

< 4;(3)

(b)

xe x 1. ( x 2 1)

.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. ..............................................................................................................................................(3) (Total 6 marks)

43

49.

The function f is defined by f (x) = cosec x + tan 2x.

(a)

Sketch the graph of f for Hence state (i) (ii) (iii) the x-intercepts;

x . 2 2

the equations of the asymptotes; the coordinates of the maximum and minimum points.(8)

(b)

Show that the roots of f (x) = 0 satisfy the equation 3 2 2 cos x 2 cos x 2 cos x + 1 = 0.(5)

(c)

Show that the x-coordinates of the maximum and minimum points on the 5 3 2 curve satisfy the equation 4 cos x 4 cos x + 2 cos x + cos x 2 = 0.(8)

(d)

Show that f ( x) + f ( + x) = 0.(4) (Total 25 marks)

44

50.

(a) Sketch the curve f (x) = | 1 + 3 sin (2x)| , for 0 x . Write down on the graph the values of the x and y intercepts.

(4)

(b)

By adding one suitable line to your sketch, find the number of solutions to the equation f (x) = ( x).

.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. ..............................................................................................................................................(2) (Total 6 marks)

45

51.

A system of equations is given by cos x + cos y = 1.2 sin x + sin y = 1.4. (a) (b) For each equation express y in terms of x.(2)

Hence solve the system for 0 < x < , 0 < y < .

(4)

.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. ..............................................................................................................................................(Total 6 marks)

46

52.

The graph of y = f (x) for 2 x 8 is shown.

1 , f ( x ) clearly showing any asymptotes and On the set of axes provided, sketch the graph of y =indicating the coordinates of any local maxima or minima.

(Total 5 marks)

47

53.

Find the set of values of x for which

0.1x 2 2 x + 3 < log 10 x.

.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. ..............................................................................................................................................(Total 6 marks)

54.

Find the values of x for which 5 3x x + 1 . Working:


48

55.

3 2 Solve the inequality x 4 + x < 0. Working:

Answer: .......................................................................... ..........................................................................(Total 6 marks)

56.

Solve the inequality 2 + 1 x 2x . Working:


49

57.

x+4 x2 Let f (x) = x + 1 , x 1 and g (x) = x 4 , x 4.Find the set of values of x such that f (x) g (x). Working:


58.

Solve the inequality

x+9 x9 2Working:


50

59.

x+4 x2 Let f (x) = x + 1 , x 1 and g (x) = x 4 , x 4. Find the set of values of x such that f (x) g (x)................................................................................................................................................ ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ...............................................................................................................................................(Total 6 marks)

51

60.

x 2 + 5x + 5 x+2 Let f (x) = , x 2.

(a) (b)

Find f (x). Solve f (x) > 2. o r k i n g :

W

A

n

s w

e r s :

( a ) ( b )(Total 6 marks)

52

function hl

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