3.14: Completed square, restricted inverse of a quadratic

0606/23/O/N/24 — Question 2 · 9 marks

The function

f

is defined by

f(x) = 1 - 4x - x^{2}

for all real values of

x

.

(a)

Write

f(x)

in the form

a - (x + b)^{2}

, where

a

and

b

are constants.

[2]

(b)

Find the range of

f

.

[1]

(c)

The function

g

is defined by

g(x) = 1 - 4x - x^{2}

for

x \geqslant k

, where

k

is a constant.

State the least possible value of

k

such that

g

has an inverse.

[1]

(d)

Using your value of

k

, find

g^{-1}(x)

, stating its domain and range.

[5]