Composition of 3 functions is even or no.

Question

If $f,g,h$ are functions defined on $\mathbb{R},$ the function $(f\circ g \circ h)$ is even:

i) If $f$ is even.

ii) If $g$ is even.

iii) If $h$ is even.

iv) Only if all the functions $f,g,h$ are even.

Shall I take different examples of functions and see?

Could anyone explain this for me please?

Answer 1

Consider the simpler case of two functions $f$ and $g$ and note that if $g$ is even then $f(g(-x))=f(g(x))$. Can you take it from here?

P.S. In order to show that a proposition is true we need a proof otherwise a counter-example suffices.

Answer 2

Only $h$ has to be even. Then $h(x)=h(-x)$ for all $x$ and $ f(g(h(x)))=f(g(h(-x)))$ for all $x$.