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)A.I onlyB.II onlyC.I and II onlyD.II and III onlyE.

To determine which of the given expressions must be even, we need to analyze each one based on 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):

   • For $f(x) + g(x)$ to be even, $(f(x) + g(x)) = (f(-x) + g(-x))$.
   • 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 equal to $f(x) + g(x)$, so $f(x) + g(x)$ is not even.

3. f(x)g(x):

   • For $f(x)g(x)$ to be even, $(f(x)g(x)) = (f(-x)g(-x))$.
   • Since $f(x)$ is even, $f(-x) = f(x)$.
   • Since $g(x)$ is odd, $g(-x) = -g(x)$.
   • Therefore,