3.5: Completed-square range and inverse of a square-root quadratic

0606/23/M/J/21 — Question 9 · 10 marks

(a)(i)

The function

f

is defined, for all real

x

, by

f(x) = 13 - 4x - 2x^{2}

.

Write

f(x)

in the form

a + b(x + c)^{2}

, where

a

,

b

and

c

are constants.

[3]

(a)(ii)

Hence write down the range of

f

.

[1]

(b)(i)

The function

g

is defined, for

x \geqslant 1

, by

g(x) = \sqrt{x^{2} + 2x - 1}

.

Given that

g^{-1}(x)

exists, write down the domain and range of

g^{-1}

.

[2]

(b)(ii)

Show that

g^{-1}(x) = -1 + \sqrt{px^{2} + q}

, where

p

and

q

are integers.

[4]