Question

Suppose that f is even and g is odd.
a. Show that f o g is even.
B. show that g o f is even .

Answer 1

Hey Conad :)

If f(x) is an even function, it satisfies this property: f(-x) = f(x).
If g(x) is an odd function, it satisfies this property: g(-x) = -g(x).

So what happens when you take their composition.
f(g(x)) = ?
Well again, let's plug in the negative x and see what happens to the whole thing:
=f(g(-x))
Since g is an odd function, g(-x) will become -g(x).
=f(-g(x))
Since f is an even function, f(-stuff) will become f(stuff).
=f(g(x))

So we just showed that f(g(-x)) = f(-g(x)) = f(g(x)).
The same property that even functions have.