Exercise:
Be $f,g: \mathbb{R}\to\mathbb{R}$ two functions. $f$ is odd, and $g$ is even. Prove that $f(g(x))$ is odd, and $g(f(x))$ is even.
I personally think there is a mistake since both the compositions should be even. Indeed through $x \to -x$:
$$f(g(x)) \to f(g(-x)) = f(g(x))$$
$$g(f(x)) \to g(f(-x)) = g(-f(x)) = g(f(x))$$
Am I right or is there something I'm missing?