2.7: Linear-quadratic composite and surd inverse

0606/22/M/J/19 — Question 12 · 9 marks

(a)(i)

The functions

f

and

g

are defined by

f(x) = 5x - 2

for

x > 1

and

g(x) = 4x^{2} - 9

for

x > 0

State the range of

g

[1]

(a)(ii)

Find the domain of

gf

[1]

(a)(iii)

Showing all your working, find the exact solutions of

gf(x) = 4

[3]

(b)(i)

The function

h

is defined by

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

for

x \leqslant -1

State the geometrical relationship between the graphs of

y = h(x)

and

y = h^{-1}(x)

[1]

(b)(ii)

Find an expression for

h^{-1}(x)

[3]