Why does the function f(|x|) look like this? If the image is f(x), why does f(|x|) look like two triangles above the x axis (basically the right side duplicated on the left)?
Ok, so let us call $g(x)=f(|x|)$. You agree that $g(x)=f(x)$ if $x\geq 0$, so the graph remains unchanged. 

What happens when you try to calculate say $g(-1)$?. You get $f(|-1|)=f(1)=2$, it is the same for all $x$ values that are negative, so that is why you get a copy of the triangle.
Ok I think I should have put it in a better way like the other answers. Let $g(x)=f(|x|)$ and now $g(x)$ will look like the positive part of $f(x)$ flipped if $x&lt;0$ since $$g(x)=\left\{ \begin{array}{ll} f(x) &amp; x\geq0 \\ f(-x) &amp; x&lt;0\end{array}\right.$$
Now, $g(-2)=f(2)$ and so forth.