Practice section 2 of 2

It is given that $f(x)=\begin{cases}3+x&0<x\le2,\\7-x&2<x\le4,\end{cases}$ and $f(x)=f(x+4)$ for all real $x$. (i) Find $f(1)$, $f(8)$ and $f(29)$. (ii) Sketch $y=f(x)$ for $-3<x\le8$.

Find the range of each of the following functions.

(a) $f:x\mapsto x^2+4x+3$, $x\in\mathbb R$, $x\leq-2$.

(b) $g:x\mapsto e^{2x}$, $x\in\mathbb R$, $x>0$.

(c) $h:x\mapsto\sin x$, $x\in\mathbb R$, $-\pi\leq x\leq\pi$.

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 $-2<x\leq8$.

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]

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

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]

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