I need to find the formula for h(x) for all x

Question

we're given a function $h(x)=\begin{cases}1/2&\text{for}&0\le x<2\\0 &\text{otherwise}\end{cases}$.

Then we are told to define the function $\displaystyle g(x)= \int_{-\infty}^{+\infty} h(y)h(x-y)dy$ I need to find the formula for $h(x)$ for all $x$.

what i believe i need to do is a double integral with integral from $-\infty$ to $+\infty$ as outside integral and integral from $0$ to $2$ of $\frac{1}{2}h(x-y)$

i don't know how to handle the $h(x-y)$ part

Answer 1

Hint: $h(y) h(x-y) = 1/4$ if ... and $0$ otherwise. Draw a picture of this region in the $xy$ plane.

Answer 2

Note that the integral is over $y$ and $x$ is a constant, so obviously the domain for $h(y)$ is $[0,2]$, but $x-y$ might not be in $[0,2]$ and then it is equal to $0$, therefore, you need to find out for which $x$, you have $y$'s that $x-y$ is also in $[0,2]$ when $y$ is in that interval.