1.11: Identifying composites and rational function

0606/12/O/N/19 — Question 5 · 7 marks

(a)(i)

It is given that

f : x \mapsto \sqrt{x}

for

x \geqslant 0

and

g : x \mapsto x + 5

for

x \geqslant 0

.

Identify each of the following functions with one of

f^{-1}

,

g^{-1}

,

fg

,

gf

,

f^{2}

,

g^{2}

.

\sqrt{x + 5}

[1]

(a)(ii)

x - 5

[1]

(a)(iii)

x^{2}

[1]

(a)(iv)

x + 10

[1]

(b)(i)

It is given that

h(x) = a + \dfrac{b}{x^{2}}

where

a

and

b

are constants.

Why is

-2 \leqslant x \leqslant 2

not a suitable domain for

h(x)

?

[1]

(b)(ii)

Given that

h(1) = 4

and

h'(1) = 16

, find the value of

a

and of

b

.

[2]