3.1: Inverse of a radical quotient and composite with exponential

0606/22/F/M/21 — Question 10 · 8 marks

3.1 diagram

The function

f

is defined by

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

for

0.5 \leqslant x \leqslant 1.5

.

The diagram shows a sketch of

y = f(x)

.

(a)(i)

It is given that

f^{-1}

exists. Find the domain and range of

f^{-1}

.

[3]

(a)(ii)

Find an expression for

f^{-1}(x)

.

[3]

(b)

The function

g

is defined by

g(x) = \mathrm{e}^{x^{2}}

for all real

x

. Show that

gf(x) = \mathrm{e}^{\left(1 - \dfrac{a}{bx^{2}}\right)}

, where

a

and

b

are integers.

[2]