Discussion questions

Determine the range of each of the following functions. State, with reasons, whether the inverse function exists in each case. Then, find the rule and domain of the inverse function if it exists.

(a) $h:x\mapsto\cos(x+1)$, $-1\leq x\leq-1+\frac{\pi}{2}$.

(b) $q:x\mapsto e^x(x-1)$, $x\in\mathbb R$.

The function $f$ is defined by $f:x\mapsto(x-4)^2+1$ for $x\in\mathbb R$, $x>4$.

(i) Sketch the graph of $y=f(x)$. Your sketch should indicate the position of the graph in relation to the origin.

(ii) Find $f^{-1}(x)$, stating the domain of $f^{-1}$.

(iii) On the same diagram as in part (i), sketch the graph of $y=f^{-1}(x)$.

(iv) Write down the equation of the line in which the graph of $y=f(x)$ must be reflected in order to obtain the graph of $y=f^{-1}(x)$, and hence find the exact solution of the equation $f(x)=f^{-1}(x)$.

Functions $f$, $g$ and $h$ are defined by

$f:x\mapsto|x^2-6x+5|$ for $x\in\mathbb R$, $2\leq x\leq6$,

$g:x\mapsto\ln(x^2)$ for $x\in\mathbb R$, $x\ne0$,

$h:x\mapsto\dfrac1{2-x}$ for $x\in\mathbb R$, $x\leq\dfrac32$.

(i) Give a reason why $f$ does not have an inverse.

(ii) State the longest possible interval in the domain of $f$, to which the restriction of $f$ would have an inverse. Hence, define $f^{-1}$ in similar form for function $f$ restricted to this interval.

(iii) Show that the composite function $gh$ exists.

(iv) Determine the exact range of function $gh$.

The function $f$ is defined by $f:x\mapsto\dfrac{ax}{bx-a}$ for $x\in\mathbb R$, $x\ne\dfrac ab$, where $a$ and $b$ are non-zero constants.

(i) Find $f^{-1}(x)$. Hence or otherwise find $f^2(x)$ and state the range of $f^2$.

(ii) The function $g$ is defined by $g:x\mapsto\dfrac1x$ for all real non-zero $x$.

State whether the composite function $fg$ exists, justifying your answer.

(iii) Solve the equation $f^{-1}(x)=x$.

It is given that

$g(x)=\begin{cases}3,&0<x\leq2,\\5-x,&2<x\leq4,\end{cases}$

and that $g(x)=g(x+4)$ for all real values of $x$.

(i) Find $g(3)$, $g(8)$ and $g(101)$.

(ii) Sketch the graph of $y=g(x)$ for $-2<x\leq8$.

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 $fg$.