### Instruction: Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$.  Find $g^{-1}(f(-7))$.### Response: To solve this problem, I need to use the idea that inverse functions undo each other.
That is, if $f^{-1}(g(x))=5x+3$, then applying $f$ to both sides gives $g(x)=f(5x+3)$.
Similarly, if I apply $g^{-1}$ to both sides, I get $x=g^{-1}(f(5x+3))$.
This is a useful equation, because it relates $x$ and $g^{-1}(f(x))$.
Now, I want to find $g^{-1}(f(-7))$, so I need to figure out what value of $x$ makes $5x+3=-7$.
Solving for $x$, I get $x=-2$.
So, $g^{-1}(f(-7))=g^{-1}(f(5(-2)+3))$.