Practice section 2 of 2

The function $f$ is defined by

$f:x\mapsto\begin{cases}\cos(\pi x/2),&0\leq x<1,\\1-x,&1\leq x\leq2.\end{cases}$

(i) Sketch the graph of $f$, indicating clearly the coordinates of the axial intercepts and endpoints.

(ii) Show that $f$ has an inverse, and define it in similar form.

(iii) Find the range of values of $x$ for which $ff^{-1}(x)=f^{-1}f(x)$.

(iv) The function $g$ is defined by $g:x\mapsto e^x$, $0\leq x\leq\ln2$.

Find the range of $fg$.

The functions $f$ and $g$ are defined by

$f:x\mapsto\sin\left(x+\dfrac\pi4\right)$, $x\in\mathbb R$, $-\dfrac\pi2\leq x\leq\dfrac\pi2$,

$g:x\mapsto\ln(2-x)$, $x\in\mathbb R$, $x<2$.

Explain whether the following composite function exist

(a) $gf$

(b) $fg$

If the composite function exists, determine the rule, domain and range. [5]

(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]

(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]

(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]