Finding the Inverse of a Function: Solving for y=x+√(x^2-1)

[anthonych414]

Homework Statement

I need to find the inverse of y=x+√(x^2-1)

Homework Equations

The Attempt at a Solution

I know it's undefined from x=-1 and x=1 so there must be two different inverse functions on two different intervals. I don't know how to find them though.

 

[HallsofIvy]

You find the inverse pretty much the way you find any inverse:
Given $y= x+ \sqrt{x^2- 1}$, solve for x. Since there is a square root, we will want to square, and, in order not to get another square root in the "cross term" we want it by itself: $y- x= \sqrt{x^2- 1}$. Now square that and solve for x.

Note the while x cannot be between -1 and 1, y can go to 0.

 

[SammyS]

Homework Statement

I need to find the inverse of y=x+√(x^2-1)

Homework Equations

The Attempt at a Solution

I know it's undefined from x=-1 and x=1 so there must be two different inverse functions on two different intervals. I don't know how to find them though.

Let's call the function you are given, f, so that

$f(x)=x+\sqrt{x^2-1}\ .$​

Following HallsofIvy's suggestion you will find:

$x=g(y)\ .$​

For the function, g, to be the inverse of function, f, it must also be true that g is the inverse of f. However, the domain of g will need to be restricted (to the range of f) so that g is a 1 to 1 function.