Question

What is the inverse of `f(x) = 2ln(x-1)-ix-x`?

Answer 1

`f^(-1)(x)=e^(ix)-x+1`#

Explanation:

If f"`f(x)=2"ln"(x+1)-ix-x`, then `f^(-1)(x)`is the inverse, or the reflection of `f(x)` in the line `y=x`.

So, let's make `f(x)=2"ln"(x+1)-ix-x=0`, then `2"ln"(x+1)=ix+x=i(2x)`, dividing both sides by 2 gives `f(x)="ln"(x+1)=(i(2x))/2=ix`.

As `e^("ln"(a))=a`, `e^(ln(x+1))=x+1=e^(ix)`.

Now take away `(x+1)` from both sides to make it equal 0, `0=e^(ix)-x+1`.