Answer
$f+g$ is a linear function.
$f g$ is not linear.
Work Step by Step
Let $f(x)=ax+b, g(x)=cx+d$ be two linear functions. Then, the sum is
$(f+g)(x)= (a+c) x + (b+d)$
which is still in a linear form; that is, $f+g$ is a linear function.
But for the product $f g$ of $f$ and $g$ is:
$(f g)(x) = (ac)(x^2) + (ad+bc) (x) + (b d)$
which is a second-order degree and not linear. Thus, $f g$ is not linear.
