3.23: Function notation table and inverse of $h$

0606/13/M/J/22 — Question 6 · 10 marks

It is given that

\mathrm{f}\colon x \mapsto 2x^{2}

for

x \geqslant 0

and

\mathrm{g}\colon x \mapsto 2x + 1

for

x \geqslant 0

(a)

Complete the table, writing each expression as one of

\mathrm{f}

\mathrm{f}^{-1}

\mathrm{g}

\mathrm{g}^{-1}

\mathrm{fg}

\mathrm{gf}

\mathrm{f}^{2}

\mathrm{g}^{2}

[5]

(b)(i)

It is given that

h(x) = (x - 1)^{2} + 3

for

x \geqslant a

. The value of

a

is as small as possible such that

h^{-1}

exists.

Write down the value of

a

[1]

(b)(ii)

Write down the range of

h

[1]

(b)(iii)

Find

h^{-1}(x)

and state its domain.

[3]

3.23: Function notation table and inverse of hhh