find Inverse function. [ f(x)=sqrt(x^2)-5 ]

To find the inverse function of \( f(x) = \sqrt{x^2 - 5} \), we need to follow these steps:1. **Express in terms of :** 2. **Swap and :** 3. **Solve for :** - First, square both sides to eliminate the square root: - Then, solve for : - Finally, take the square root of both sides: 4. **Determine the domain and range:** - The original function \( f(x) = \sqrt{x^2 - 5} \) is defined for , which means or . - The range of \( f(x) \) is \( [0, \infty) \) because the square root function outputs non-negative values. Since the inverse function must be a function (i.e., it must pass the vertical line test), we restrict the domain of the original function to . This ensures that the original function is one-to-one and thus has an inverse.5. **Write the inverse function:** Given the restriction , the inverse function is: So, the inverse function of \( f(x) = \sqrt{x^2 - 5} \) is: