Question

FAN AND MEDAL !! Let f(x) be even and g(x) be odd . Using the properties of even and odd functions , is f(g(x)) and g(f(x)) even, odd, or neither ?

Answer 1

try an example ... f(x)=x^2 and g(x) = x

Answer 2

An even function \(g\): means that \(g(−x)=g(x)\)
An off function \(m\): means that \(m(−x)=−m(x)\)

To test if the composition \(f(g(x))\) is even, you should check that
\((f∘g)(−x)=(f∘g)(x)\)

So:
\((f∘g)(−x)=f(g(−x))=f(−g(x)) \)since \(g\) is odd.
\(f(−g(x))=f(g(x))=(f∘g)(x) \) since\( f\) is even

Can you try the same idea for the other composition?