Find Inverse of f(x)=\frac{7e^{x}-6}{e^{x}+8}: Solve x

[7yler]

Find a formula for the inverse of f.

f(x)=\frac{7e^{x}-6}{e^{x}+8}

I set f(x) equal to y, and operated on it until I got:

ln(y)=ln(7e^{x}-6)-ln(e^{x}+8)

But I'm stuck. I'm not sure how to isolate x.

 

[Mark44]

Find a formula for the inverse of f.

f(x)=\frac{7e^{x}-6}{e^{x}+8}

I set f(x) equal to y, and operated on it until I got:

ln(y)=ln(7e^{x}-6)-ln(e^{x}+8)

But I'm stuck. I'm not sure how to isolate x.

y = \frac{7e^{x}-6}{e^{x}+8}
Multiply both sides by e^x + 8 to get

y(e^x + 8) = 7e^{x}-6

Now expand the left side. Then bring all the terms that involve x to the left side, and move all the other terms to the right side.
Solve for x.

The basic idea is that if y = f(x), and f is invertible, then x = f^-1(y).

That gives you the inverse function, as a function of y. To write the inverse as a function of x, simply change each occurrence of y in the formula of the inverse to an x.

 

[7yler]

That was a huge help. Thank you.

 

[epenguin]

You might simplify it before you invert rather than after . I think the RHS is

7 - \frac{62}{e^x + 8}

Something like that anyway.