How to Find the Domain of The Inverse of (4(e^x)-5)/(25(e^x)+12)

Question

These are the steps I have taken so far:

In order to find the inverse of the function, I did the following steps:

1) Make all (e^x) equal to y, so that the equation is (4y-5)/(25y+12)
2) Get 'y' by itself, so that y = (-5-12(ln(e^x)))/(25(ln(e^x))-4)

Now, to get the domain, I realized that the denominator cannot be equal to 0, so that (ln(e^x)) cannot equal 4/25. As well, I realized that a logarithm's base must be greater than 0. However, I am not sure how to factor this into calculating the domain.

Does anyone know the steps to getting this domain?

All help is appreciated.

Answer 1

Hint:

1) Let $u=e^x$, and then solve the equation $\displaystyle y=\frac{4u-5}{25u+12}$ for $u$ to get an equation of the form $u=f(y)$.

2) Substitute $e^x$ back for $u$ and solve for $x$ to get $x=\ln\big(f(y)\big)$.

3) Now solve $f(y)>0$ to find the domain of the inverse function.