Answer
False.
Work Step by Step
Let's test whether a value $-1$ is in the range. If $-1$ is in the range, there exists an $x$ such that $f(x)=-1$. So, $$4-\frac{1}{x}=-1$$ Subtract $4$ from both sides to get $$-\frac{1}{x}=-5$$ Take the reciprocal of both sides to get $$-x=-\frac{1}{5}$$ Therefore, $$x=\frac{1}{5}$$ Since $f(\frac{1}{5})=-1$, $-1$ is in the range. However, $-1$ isn't in the interval $(0,4)$. Thus, the statement must be false.
