Find the inverse of the function $f(x) = \sqrt{x^2 - 1}$.

1. Problem Description

Find the inverse of the function $f(x) = \sqrt{x^2 - 1}$.

2. Solution Steps

To find the inverse of the function $f(x) = \sqrt{x^2 - 1}$, we follow these steps:

Step 1: Replace $f(x)$ with $y$.

$y = \sqrt{x^2 - 1}$

Step 2: Swap $x$ and $y$.

$x = \sqrt{y^2 - 1}$

Step 3: Solve for $y$.

Square both sides:

$x^2 = y^2 - 1$

Add 1 to both sides:

$x^2 + 1 = y^2$

Take the square root of both sides:

$y = \pm \sqrt{x^2 + 1}$

Step 4: Replace $y$ with $f^{-1}(x)$.

$f^{-1}(x) = \pm \sqrt{x^2 + 1}$

Since the original function has $x^2$ inside the square root, we must consider the domain of the original function. The domain is $x \le -1$ or $x \ge 1$. The range of the original function is $y \ge 0$. Thus, the domain of the inverse is $x \ge 0$. Because $x \ge 1$ in the domain of the original function, we take the positive solution: $f^{-1}(x) = \sqrt{x^2 + 1}$.

3. Final Answer

$f^{-1}(x) = \sqrt{x^2 + 1}$