Given two functions $f(x)$ and $g(x)$,
$h(x) = f(x)-g(x)$
and
$i(x) = f(x) + g(x)$
can you know how many times $h(x)$ and $i(x)$ will intersect?
I ask this because I noticed that if $f(x)$ and $g(x)$ are linear, I only observed one point of intersection between $h(x)$ and $i(x)$.
When $f(x)$ and $g(x)$ are quadratic, I noticed two points of intersection between $h(x)$ and $i(x)$.
It seems like the number of intersections is related to the power of the functions $f(x)$ and $g(x)$ from which $h(x)$ and $i(x)$ are constructed from but I'm not sure how to prove it. Is there a way to do this?
$$f(x)+g(x)=f(x)-g(x) \to 2g(x)=0 \to g(x)=0$$