So I have two functions with me:
f(x) = 2*x -1 and g(x) = 1/2*x+4
We were first asked to determine the following: f*g(x). Solving it, my solution was: x^2+ 7.5*x - 4
It was then asked to determine (f*g)^-1. Working with my previous solution I notice that it's quite complicated to find the inverse. I'm wondering if I went wrong somewhere. Plugging this equation in to Wolfram Alpha, I get an inverse of: -3.75±0.25 sqrt(16. x+289.)
This seems like a rather complicated answer for a relatively simple 1st semester calculus course.
I may try to generalize. Assuming that both $f$ and $g$ are invertible, we have $$f\circ g(x) = f(g(x)) = y$$ $$f^{-1}(f(g(x))) = f^{-1}(y)$$ $$g(x) = f^{-1}(y)$$ $$g^{-1}(g(x)) = g^{-1}(f^{-1}(y))$$ $$x = g^{-1}(f^{-1}(y))$$
Now just find the inverses of $f$ and $g$, and use that information to compose $f^{-1}\circ g^{-1}$