Mathematics AA SL

FUNCTIONS
Worksheet 1

1. No GDC
Let f(x) = 7 – 2x and g(x) = x + 3.
(a) Find (g ° f)(x).
(2)
(b) Write down g–1(x).
(1)
–1
(c) Find (f ° g )(5).
(2)
(Total 5 marks)

2. No GDC
Consider f(x) = 2kx2 – 4kx + 1, for k ≠ 0. The equation f(x) = 0 has two equal roots.
(a) Find the value of k.
(5)
(b) The line y = p intersects the graph of f. Find all possible values of p.
(2)
(Total 7 marks)

3. No GDC
Let f(x) = 3 ln x and g(x) = ln 5x3.
(a) Express g(x) in the form f(x) + ln a, where a  +
.
(4)
(b) The graph of g is a transformation of the graph of f. Give a full geometric description of
this transformation.
(3)
(Total 7 marks)

4. No GDC
The following diagram shows part of the graph of a quadratic function f.

The x-intercepts are at (–4, 0) and (6, 0) and the y-intercept is at (0, 240).

(a) Write down f(x) in the form f(x) = –10(x – p)(x – q).
(2)
2
(b) Find another expression for f(x) in the form f(x) = –10(x – h) + k.
(4)
2
(c) Show that f(x) can also be written in the form f(x) = 240 + 20x – 10x .
(2)
–1
A particle moves along a straight line so that its velocity, v m s , at time t seconds is given by
v = 240 + 20t – 10t2, for 0 ≤ t ≤ 6.
 (d) (i) Find the value of t when the speed of the particle is greatest.
(ii) Find the acceleration of the particle when its speed is zero.
(7)
(Total 15 marks)

5. No GDC
Let f(x) = 3x2. The graph of f is translated 1 unit to the right and 2 units down.
The graph of g is the image of the graph of f after this translation.
(a) Write down the coordinates of the vertex of the graph of g.
(2)
2
(b) Express g in the form g(x) = 3(x – p) + q.
(2)
The graph of h is the reflection of the graph of g in the x-axis.
(c) Write down the coordinates of the vertex of the graph of h.
(2)
(Total 6 marks)

6. No GDC
x
Let f(x) = log3 + log3 16 – log3 4, for x > 0.
2
(a) Show that f(x) = log3 2x.
(2)
(b) Find the value of f(0.5) and of f(4.5).
(3)
ln ax
The function f can also be written in the form f(x) = .
ln b
(c) (i) Write down the value of a and of b.

(ii) Hence on graph paper, sketch the graph of f, for –5 ≤ x ≤ 5, –5 ≤ y ≤ 5, using a

scale of 1 cm to 1 unit on each axis.

(iii) Write down the equation of the asymptote.

(6)
–1
(d) Write down the value of f (0).
(1)
The point A lies on the graph of f. At A, x = 4.5.
(e) On your diagram, sketch the graph of f–1, noting clearly the image of point A.
(4)
(Total 16 marks)

7. No GDC
Let f(x) = 3x, g(x) = 2x – 5 and h(x) = (f ° g)(x).
(a) Find h(x).
(2)
–1
(b) Find h (x).
(3)
(Total 5 marks)
 1
8. Let g(x) = x sin x, for 0 ≤ x ≤ 4.
2
(a) Sketch the graph of g on the following set of axes.

(4)
(b) Hence find the value of x for which g(x) = –1.
(2)
(Total 6 marks)

9. No GDC
Let f(x) = 8x – 2x2. Part of the graph of f is shown below.

(a) Find the x-intercepts of the graph.

(4)
(b) (i) Write down the equation of the axis of symmetry.
(ii) Find the y-coordinate of the vertex.
(3)
(Total 7 marks)

10. No GDC
Let f(x) = log3 x , for x > 0.
(a) Show that f–1(x) = 32x.
(2)
–1
(b) Write down the range of f .
(1)
Let g(x) = log3 x, for x > 0.
(c) Find the value of (f –1 ° g)(2), giving your answer as an integer.
(4)
(Total 7 marks)