Mathematics AA HL
Functions
Worksheet 6
51. No GDC
Find the range of values of m such that for all x
m(x + 1) x2.
(Total 6 marks)
52. No GDC
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
10
2 P(1, 1)
×
–10 –8 –6 –4 –2 0 2 4 6 8 10 x
–2
–4
–6
–8
–10
Find the exact coordinates of Q and of R.
(Total 6 marks)
53. No GDC
The polynomial f(x) = x3 + 3x2 +ax + b leaves the same remainder when divided by (x –
2) as when divided by (x +1). Find the value of a.
(Total 6 marks)
54. No GDC
The functions f and g are defined by f : x ex, g : x x + 2.
(a) Calculate f–1(3) × g–1(3).
(b) Show that (f ° g)–1(3) = ln 3 – 2.
(Total 6 marks)
x+4 x−2
55. Let f(x) = , x ≠ –1 and g(x) = , x ≠ 4. Find the set of values of x such that
x +1 x−4
f(x) g(x).
(Total 6 marks)
56. No GDC
Let f(x) = x + 5 x + 5 , x –2.
2
x+2
(a) Find f(x).
(b) Solve f(x) > 2.
(Total 6 marks)
57. No GDC
Let f(x) = x + 5 x + 5 , x –2.
2
x+2
(a) Find f(x).
(b) Solve f(x) > 2.
(Total 6 marks)
58. No GDC
The function f is defined for x > 2 by f(x) = ln x + ln (x – 2) – ln (x2 – 4).
(a) Express f(x) in the form ln x .
x+a
(b) Find an expression for f–1(x).
(Total 6 marks)
