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?
I really don't know how to approach this :/ I'm really bad. (Sorry I'm just starting 6th grade so I'm quite a beginner at competitive maths.) Any solutions? Thanks.
Since $f$ is one-to-one, $f(x^{2})=f(x^{4})$ iff $x^{2}=x^{4}$ which means $x^{2}=0$ or $x^{2}=1$. Thus, $x$ is $0,-1$ or $+1$. and these are the the values for which the graphs intersect. So the answer is $3$.