Invert an exponential and distinguish a composite from a square

(a) The functions $f$ and $g$ are defined by

$f:x\mapsto e^{1-x^2}$, $x>0$,

$g:x\mapsto\dfrac{1+x}{x}$, $x>0$.

(i) Find, in a similar form, $f^{-1}$ and $g^2$.6 marks

(ii) Sketch the graph of $g^2$ and write down the range of values of $m$ such that $g^2(x)=mx+2$ has no real roots for $x>0$.3 marks

(b) The functions $h$ and $k$, each with domain $\mathbb R$, are such that

$h:x\mapsto(x-1)^2+5$,

$kh:x\mapsto2x^2-4x+15$.

Find the function $k$.2 marks