Suppose $f(x)$ is a function defined for all real $x$, and suppose $f$ is invertible (that is, $f^{-1}(x)$ exists for all $x$ in the range of $f$).

If the graphs of $y=f(x^2)$ and $y=f(x^4)$ are drawn, at how many points do they intersect?
The graphs of $y=f(x^2)$ and $y=f(x^4)$ intersect when $f(x^2)=f(x^4)$. Since $f$ is invertible, this means that $x^2=x^4$, or equivalently, $x^4-x^2=0$. I can factor this equation as $x^2(x^2-1)=0$. So the solutions are $x=0$, $x=1$, and $x=-1$. These are the only values of $x$ for which $f(x^2)=f(x^4)$, so the graphs intersect at exactly three points. 