Mathematics AA HL
Functions
Worksheet 4
31. No GDC
x 2 – 5x – 4
Find the equations of all the asymptotes of the graph of y = .
x 2 – 5x + 4
(Total 6 marks)
ln (2 ), giving the answer in the form a ln 2, where a
50
r
32. Find .
r =1
(Total 6 marks)
3
33. Solve the inequality x2 – 4 + < 0.
x
(Total 6 marks)
34. (a) On the same axes sketch the graphs of the functions, f(x) and g(x), where
f(x) = 4 – (1 – x)2, for – 2 x 4,
g(x) = ln (x + 3) – 2, for – 3 x 5.
(2)
(b) (i) Write down the equation of any vertical asymptotes.
(ii) State the x-intercept and y-intercept of g(x).
(3)
(c) Find the values of x for which f(x) = g(x).
(2)
(d) Let A be the region where f(x) g(x) and x 0.
(i) On your graph shade the region A.
(ii) Write down an integral that represents the area of A.
(iii) Evaluate this integral.
(4)
(e) In the region A find the maximum vertical distance between f(x) and g(x).
(3)
(Total 14 marks)
35. No GDC
The polynomial x3 + ax2 – 3x + b is divisible by (x – 2) and has a remainder 6 when
divided by (x + 1). Find the value of a and of b.
(Total 6 marks)
36. The function f is given by f(x) = 2 – x2 – ex.
Write down
(a) the maximum value of f(x);
(b) the two roots of the equation f(x) = 0.
(Total 6 marks)
37. Solve the inequality x – 2 2x + 1.
(Total 6 marks)
38. No GDC
x2 – 1
The function f is defined for x 0 by f(x) = .
x2 +1
Find an expression for f–1(x).
(Total 6 marks)
39. No GDC
The diagram shows the graph of f(x).
1
(a) On the same diagram, sketch the graph of , indicating clearly any
f ( x)
asymptotes.
y
–2 –1 0 1 2 x
–1
–2
(b) On the diagram write down the coordinates of the local maximum point, the local
1
minimum point, the x-intercepts and the y-intercept of .
f ( x)
(Total 6 marks)
40. Consider the equation (1 + 2k)x2 – 10x + k – 2 = 0, k . Find the set of values of k
for which the equation has real roots.
(Total 6 marks)
