If f(x) is an even function and g(x) is an odd function, which of the following must be even?I. f(g(x))II. f(x) + g(x)III. f(x)g(x)

To determine which of the given expressions must be even, we need to use the properties of even and odd functions.

1. f(g(x)):

   • Since $g(x)$ is an odd function, $g(-x) = -g(x)$.
   • Since $f(x)$ is an even function, $f(-x) = f(x)$.
   • Therefore, $f(g(-x)) = f(-g(x)) = f(g(x))$.
   • This shows that $f(g(x))$ is even.

2. f(x) + g(x):

   • Since $f(x)$ is even, $f(-x) = f(x)$.
   • Since $g(x)$ is odd, $g(-x) = -g(x)$.
   • Therefore, $f(-x) + g(-x) = f(x) - g(x)$.
   • This is not necessarily equal to $f(x) + g(x)$, so $f(x) + g(x)$ is not necessarily even.

3. f(x)g(x):

   • Since $f(x)$ is even, $f(-x) = f(x)$.
   • Since $g(x)$ is odd, $g(-x) = -g(x)$.
   • Therefore, $f(-x)g(-x) = f(x)(-g(x)) = -f(x)g(x)$.
   • This shows that $f(x)g(x)$ is odd, not even.

So, the only expression that must be even is:

I. $f(g(x))$