Haef Ib - Math Hl

HAEF IB - MATH HL TEST 2 FUNCTIONS by Christos Nikolaidis Marks:____/100 Name:______

Date:______Grade: ______

Questions

1. [Maximum mark: 7]

Let f (x) = x 1 1 and g(x) = x 2 . [4 marks] (a) Solve the equation (gof )(x) = 1 [3 marks] (b) Find the function h(x) given that hof = g

......

...... Page 1 ......

......

2. [Maximum mark: 5]

Find a point A on the line y  2x and a point B on the parabola y  x 2  2 which lies in the first quadrant, so that M(3,11) is the midpoint of the line segment [AB]. [5 marks]

......

Page 2 ......

......

3. [Maximum mark: 7]

The polynomial f (x)  x 3  8x 2  ax  b is divisible by (x 1) and has a remainder

(x 1) –36 when divided by . [4 marks]

(a) Find the value of a and of b . [1 mark] (b) Find the remainder when f (x) is divided by (x  2) (c) Solve the equation f (x)  0 [2 marks]

......

...... Page 3 ......

......

Page 4 4. [Maximum mark: 9]

The functions f and g are both defined in the interval [-4,4] and g is invertible.

Some values of the functions are given below.

x 1 3 4 x 1 3 4 f (x) 3 2 1 g(x) 4 1 -3

[2 marks] (a) Calculate (g 1 ° f )(4) (b) Find a solution of the equation ( f ° g)(x)  1 [2 marks] [1 mark] (c) Calculate (g 1 ° g)(2)

[2 marks] Given that the function f is even and the function g is odd (d) Calculate ( f ° g)(4) . [2 marks]

(e) Find a solution of the equation ( f ° g)(x)  2

......

Page 5 ......

......

5. [Maximum mark: 13]

3x  6 Consider the function f (x) = x  4

(a) Complete the following table

Horizontal Vertical Function x-intercept y-intercept Asymptote Asymptote y  f (x)

1 y  f (x)

y  f 1 (x) [6 marks]

(b) Sketch the graph of f (x) by indicating any asymptotes and intersections with x- and y-axes.

Page 6 y 6

1 x -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 -1

-2

-3

-4 [3 marks]

(c) Let A be the point of intersection of the two asymptotes of f (x) . Find the corresponding position to the point A under the following transformations:

Transformation f (x)+2 f (x–7) – 2f (x) f (2x–4) Corresponding point [4 marks]

Page 7 6. [Maximum mark: 5]

The diagram shows the graphs of the functions y  f (x) and y  g(x) .

y g(x) 4 y  g(x) 3

1 x 1 2 3 4 5 6 7 8 9 -1

-2 f (x)

-3 y  g(x)

-4

g(x) Sketch the graph of y  . f (x)

Indicate clearly where the y-intercept, the x-intercepts and any asymptotes occur.

y 4

1 x 1 2 3 4 5 6 7 8 9 -1

-2

-3

-4

Page 8 Page 9 7. [Maximum mark: 7]

The function f is defined by

f (x)  1 x , for x  0

1 (a) Find an expression for f (x) . [2 marks]

(b) State the domain and the range of f 1 (x) . [2 marks] [3 marks] (c) Find the exact root of the equation f (x)  f 1 (x) .

......

Page 10 ......

......

Page 11 8. [Maximum mark: 19]

1 Let f (x) = (x  2)(x  4) . 2 (a) Express the function f (x) in the form a(x  h) 2  k .

......

...... [3 marks] ......

1 (b) Sketch the graphs of f (x) and on the same axes by indicating any possible f (x) intercepts, roots, minimum/maximum values and asymptotes.

y 4

1 x -4 -3 -2 -1 1 2 3 4 5 6 -1

-2

-3

-4

-5 [6 marks]

Page 12 1 Function y  f (x) y  y   f (x) y  f (x) y  f ( x ) f (x) Range [5 marks]

(d) Describe a sequence of transformations that maps the graph of y  x 2 to the graph of f (x)

......

[2 marks] ......

(e) If f (x) is restricted in the interval x  1, explain why f 1 (x) exists and find its expression.

......

...... Page 13 ......

......

...... [3 marks] ......

Page 14 9. [Maximum mark: 13]

3x 1 The function f (x) is defined by f (x)  , x  3. x  3 [3 marks] (a) Show that f is a self-inverse function.

( f ° f )(k) (b) Hence find, in terms of k, the result of the composite function , [1 mark] where k  3

The figure below shows a sketch of a one-to-one function g(x) defined over the domain  2  x  2 . The graph of y  g(x) consists of two straight line segments and the range of g(x) is  5  g(x)  6 .

[2 marks] (c) Find the value of ( f ° g)(2) .

(d) On the same diagram above, sketch the graph of the inverse function [3 marks]

y  g 1 (x) and state its domain.

 1  [4 marks] The function h(x) is defined by h(x)  2g (x 1) .  2 

(e) Sketch the graph of the function y  h(x) and state its range.

Page 15 ......

......

...... Page 16 ......

......

Page 17 ......

......

Page 18 ......

PART B (only for this question you may use GDC)

10. [Maximum mark: 15]

The function y  f (x) is defined as

2x 2 12x 16 f (x)  x 2  3x  4

(a) Sketch the graph of y  f (x) , clearly indicating the coordinates of all intersections with the axes, the equations of all asymptotes (vertical and horizontal) and show the approximate positions of the points on the graph representing the local minimum and the local maximum. [6 marks] (b) State the domain and the range of y  f (x) . [3 marks] (c) The equation f (x)  p has exactly one solution. Find the possible values of p. [2 marks] (d) Using your graph, or otherwise, solve the inequality 2x 2 12x 16  x 2  3 x 2  3x  4 [4 marks]

......

Page 19 ......

......

Page 20 ......

......

Page 21