Problem $13\mathrm{.}5.$ Consider the function $y=f(x)$ with multipart definition

$f(x)=\{\begin{array}{ccc}0& if& x-1\\ 2x+2& if& -1x0\\ -x+2& if& 0x2\\ 0& if& x2\end{array}$

$($a$)$ Sketch the graph of $y=f(x).$

$($b$)$ Is $y=f(x)$ an even function? Is $y=f(x)$ an odd function? $($A function $y=f(x)$ is called even if $f(x)=f(-x)$ for all $x$ in the domain. A function $y=f(x)$ is called odd if $f(-x)=-f(x)$ for all $x$ in the domain.$)$

$($c$)$ Sketch the reflection of the graph across the $x-$axis and $y-$axis. Obtain the resulting multipart equations for these reflected curves.

$($d$)$ Sketch the vertical dilations $y=2f(x)$ and $y=\frac{1}{2}f(x).$

$($e$)$ Sketch the horizontal dilations $y=f(2x)$ and $y=f(\frac{1}{2}x).$

$($f$)$ Find a number $c>0$ so that the highest point on the graph of the vertical dilation $y=cf(x)$ has $y-$coordinate $11.$

$($g$)$ Using horizontal dilation, find a number $c>0$ so that the function values $f(\frac{x}{c})$ are non$-$zero for all $c,d>0f(\frac{1}{c}(x-d))0.-\frac{5}{2}.$

$(h)$ Using horizontal dilation, find positive numbers $c,d>0so$ that the function values $f(\frac{1}{c}(x-d))$ are non$-$zero precisely when $0.$