Transform a nonconvex problem into a convex problem using perspective function

Question

Suppose I have the problem $$ \text{minimize } f_0(x)\\ \text{subject to } tf_1(x) \leq r $$ with variables $t,x \in \mathbb{R}$ and $f_0, f_1$ are convex. The constraint is not convex, so I was thinking using a variable $y = t*x$ then I can write the constraint as $tf_1(y/t) \leq r$ which is now convex. But the problem then is that the objective is no longer convex, since it is $f_0(y/t)$.