$f+g$ 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.