What's wrong with my conjecture?

Question

I was doing math homework, and I formulated the following conjecture from one of the questions: If $f(x)$, $g(x)$ and $h(x)$ are continuous functions and the equations $f(x) = h(x)$ and $g(x) = h(x)$ both have only one root, then the equation $f(x) = g(x)$ has only one root. So can anyone find a counterexample? I can't think of one, but maybe that's because I haven't read Counterexamples in Analysis yet.

Answer 1

You can just take $f(x) = g(x) = x$ and $h(x) = 0$.

If you want a less trivial example, take $f(x) = x + \sin x$, $g(x) = x + \cos x$, and $h(x) = 0$. See it here.

For a less trivial $h$, you can take $f(x) = x \sin x$, $g(x) = x \cos x$, and $h(x) = ax$ for any $a$ such that $|a| > 1$). See it here for $a=2$.

I would like to make another example, fundamentally different from the previous ones: $f(x) = -(x+1)^2$, $g(x) = (x-1)^2$, $h(x) = -4x$. Here, your conditions are also met, but $f(x) = g(x)$ has no real solutions. See it here.