inverse of quadratic log functions

Question

Can a Log function with a quadratic have an inverse function? The specific question is to find the inverse of $$f(x) = \log_2(x^2-3x-4)$$ The function already fails the horizontal line test, but apparently there is a function of

If $$x>4, \quad f^{-1}(x) = \frac{3+ \sqrt{2^{x+2}+25}}{2}$$

If $$x<-1, \quad f^{-1}(x) = \frac{3- \sqrt{2^{x+2}+25}}{2} $$ I was able to find this by swapping the x and y around, but why does it have an inverse function?

Answer 1

That's because you did that by dividing the function to two parts which individually are invertible.

Answer 2

If $\displaystyle f(x)=\log_2(x^2-3x-4)=y, x^2-3x-4=2^y $

$\displaystyle\implies f^{-1}(y)=x=\frac{3\pm\sqrt{25+2^{y+2}}}2$

Answer 3

$$\log_2(x^2-3x-4)=y$$ $$2^y=x^2-3x-4$$ $$x^2-3x-4-2^y=0$$ $$x=\frac{3\pm\sqrt{25+2^{y+2}}}{2}$$