Why is the format of my inverse function wrong?

Question

Why is the correct answer for $h(x)=\dfrac{-2x-1}{x+5}$ given by $h^{-1}(x)=\dfrac{5x+1}{-2-x}$ instead of $h^{-1}=4x-1$? Why do I stop at $\dfrac{5x+1}{-2-x}$?

Answer 1

The functions $\frac{5x+1}{-2-x}$ and $4x-1$ are not at all the same. Try plugging some numbers in to both. There's no good way to simplify a rational function like $\frac{5x+1}{-2-x}$, unless you can factor something out of both the numerator and the denominator, and then cancel.

Answer 2

$$h(x)=y=\frac{-2x-1}{x+5}=\frac9{x+5}-2$$ $$y+2=\frac9{x+5}$$ $$x+5=\frac9{y+2}$$ $$x=\frac9{y+2}-5=\frac{-5y-1}{y+2}=\frac{5y+1}{-2-y}$$ $$\therefore h^{-1}(x)=\frac{5x+1}{-2-x}$$