Practice section 2 of 2

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 $f(x)=f^{-1}(x)$.

The function f is defined as $f:x\mapsto-x^2-x+6$, $x\in\mathbb R$.

(i) Explain why $f^{-1}$ does not exist. [1]

The function f has an inverse if its domain is restricted to $x<k$.

(ii) Find the largest value of k and hence find $f^{-1}$ in a similar form. [4]

(iii) Sketch, on the same diagram, the graphs of $f$, $f^{-1}$ and $ff^{-1}$. [3]

(iv) Hence, solve the equation $f(x)=f^{-1}(x)$, leaving your answer in exact form. [2]

The function $f$ is defined by $f:x\mapsto5-(x-2)^2$, $x\leq4$.

(a) Sketch the graph of $y=f(x)$. Hence state the range of $f$. [2]

(b) Explain why $f^{-1}$ does not exist. [1]

The function $g$ is such that $g:x\mapsto f(x)$, $x\leq k$, where $k$ is a real constant.

(c) Given that $g^{-1}$ exists, state the largest possible value of $k$. [1]

Use the largest possible value of $k$ found in part (c) to answer the remaining parts of this question.

(d) Find $g^{-1}$ in similar form. [3]

(e) The function $h$ is defined by $h:x\mapsto\dfrac{\alpha x+1}{x-2}$, $x\in\mathbb R$, $x\ne2$ where $\alpha$ is a positive constant.

(i) By finding the range of $h$ in terms of $\alpha$, determine the range of values of $\alpha$ such that $h^{-1}g$ exist. [3]

(ii) Solve $h^{-1}g(x)=-11$ for the case where $\alpha=6$. [2]

The function $f$ is defined by $f:x\mapsto(ax+b)^2$, $x\in\mathbb R$, $x\geq-\dfrac ba$ where $a$ and $b$ are positive constants.

(i) Show that $f^{-1}$ exists. [2]

(ii) State the geometrical relationship between the graphs of $f$ and $f^{-1}$. Given that the graphs of $f$ and $f^{-1}$ intersect, show that $a\leq\dfrac1{4b}$. [4]

(iii) Show that the composite function $ff$ is defined. [2]

(iv) Solve the equation $ff(2)=b^2$. [3]

The function $f$ is defined by

$f:x\mapsto\ln(x-a)$, $x\in\mathbb R$, $a<x\leq a+2$,

where $a$ is a positive constant.

(i) Show that $f^{-1}$ exists. [2]

(ii) Find $f^{-1}(x)$ and state the domain of $f^{-1}$. [2]

(iii) Find an expression for $g(x)$ for each of the following cases.

(a) $fg(x)=x$, [1]

(b) $gf(x)=x^2$. [1]

(iv) The function $h$ is defined by

$h:x\mapsto\dfrac12(x-1)(x+1)(x+5)$, $x\in\mathbb R$.

Find the range of $hf$. [2]

The function $g$ is defined by

$g:x\mapsto\begin{cases}x-1,&x\leq2,\\\sqrt{x+1},&x>2.\end{cases}$

(i) Sketch the graph of $y=g(x)$. [2]

(ii) Explain why the inverse function, $g^{-1}$, exists. Find $g^{-1}$ in the similar form. [3]

Another function $h$ is defined by

$h:x\mapsto e^{x+1}$, $x\in\mathbb R$.

(iii) Find $hg(x)$ for $x<0$. [1]

(iv) Find the range of the composite function $hg$ for $x<0$. [1]