Review practice · Section 1 of 2

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 mark

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 marks

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

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

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 marks

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 marks

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

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 mark

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 marks

(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 marks

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

(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

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 marks

(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 marks

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

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

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 marks

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

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

(a) $fg(x)=x$,1 mark

(b) $gf(x)=x^2$.1 mark

(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 marks