2.3: Quadratic maximum and logarithmic composite

0606/22/M/J/16 — Question 11 · 11 marks

(a)

A function

f

is defined, for all real

x

, by

f(x) = x - x^{2}

Find the greatest value of

f(x)

and the value of

x

for which this occurs.

[3]

(b)

The domain of

g(x) = x - x^{2}

is such that

g^{-1}(x)

exists.

Explain why

x \geqslant 1

is a suitable domain for

g(x)

[1]

(c)(i)

The functions

h

and

k

are defined by

h : x \mapsto \lg(x + 2)

for

x > -2

and

k : x \mapsto 5 + \sqrt{x - 1}

for

1 < x < 101

Find

hk(10)

[2]

(c)(ii)

Find

k^{-1}(x)

, stating its domain and range.

[5]