Review practice · Section 2 of 2

It is given that

$f(x)=\begin{cases}7-x^2&\text{for }0<x\leq2,\\2x-1&\text{for }2<x\leq4\end{cases}$

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

(i) Evaluate $f(27)+f(45)$.2 marks

(ii) Sketch the graph of $y=f(x)$ for $-7\leq x\leq10$.3 marks

It is given that

$f(x)=\begin{cases}\cos^{-1}x&\text{for }0\leq x<1,\\\dfrac\pi2x-\dfrac\pi2&\text{for }1\leq x<2\end{cases}$

and that $f(x)=f(x+2)$ for all real values of $x$.

(i) Sketch the graph given by $y=f(x)$ for $-3\leq x\leq4$.3 marks

(ii) For $0\leq x\leq4$, solve the inequality $f(x)>\dfrac\pi3$, giving your answer in the exact form.3 marks

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 marks

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

Another function $h$ is defined by

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

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

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