Question

Consider the function defined as f(x) = e^x + e^-x/2, x > 0. Find the inverse of f?

Answer 1

Assuming `f(x)=(e^x+e^-x)/2`, then

`f^-1(x)=ln(x+sqrt(x^2-1))`.

But if `f(x)=e^x+e^-x/2`, then

`f^-1(x)=ln((x+sqrt(x^2-2))/2)`

Explanation:

I'm not sure if you wrote the function correctly. You wrote

`f(x)=e^x+e^-x/2`

Did you mean to find the inverse of

`f(x)=(e^x+e^-x)/2`?

If so, start with

`f(x)=y=(e^x+e^-x)/2`

Multiply both sides by 2.

`2y=e^x+e^-x`

Multiply both sides by `e^x`.

`2ye^x=(e^x)^2+1`

Subtract `2ye^x` from both sides.

`0=(e^x)^2-2ye^x+1`

Use the quadratic formula to solve for `e^x`.

`e^x=ypmsqrt(y^2-1)`

But `e^x` and `y` are all positive so we can just write

`e^x=y+sqrt(y^2-1)`

Now take the natural log of both sides.

`x=ln(y+sqrt(y^2-1))`

Another way to say this is

`f^-1(x)=ln(x+sqrt(x^2-1))`.

If you really DID mean `f(x)=e^x+e^-x/2`, then the answer is

`f^-1(x)=ln((x+sqrt(x^2-2))/2)`

using exactly the same method as above.