3.16: Inverse function, graphs and a composite equation

0606/12/O/N/23 — Question 8 · 10 marks

It is given that

\mathrm{f}\colon x\mapsto(3x+1)^2-4 \quad\text{for }x\geqslant a,

and that

\mathrm{f}^{-1}

exists.

(a)(i)

Find the least possible value of

a

[1]

(a)(ii)

Using this value of

a

, write down the range of

\mathrm{f}

[1]

(a)(iii)

Using this value of

a

, sketch the graphs of

y=\mathrm{f}(x)

and

y=\mathrm{f}^{-1}(x)

on the axes, stating the intercepts with the coordinate axes.

[4]

(b)

It is given that

\mathrm{g}(x)=\mathrm{ln}(2x^2+5) \quad\text{for }x\geqslant0,

\mathrm{h}(x)=3x-2 \quad\text{for }x\geqslant0.

Solve the equation

\mathrm{hg}(x)=4

, giving your answer in exact form.

[3]