Find the inverse of the given square root equation. \( \begin{array}{l}\text { 4. } f(x)=2 \sqrt{x-1}+5\end{array} \)

To find the inverse of the function \( f(x) = 2 \sqrt{x - 1} + 5 \), we start by replacing \( f(x) \) with \( y \): \( y = 2 \sqrt{x - 1} + 5 \). Next, isolate the square root by subtracting 5 from both sides: \( y - 5 = 2 \sqrt{x - 1} \). Now, divide by 2: \( \frac{y - 5}{2} = \sqrt{x - 1} \). To eliminate the square root, square both sides: \( \left(\frac{y - 5}{2}\right)^2 = x - 1 \). Finally, solve for \( x \): \( x = \left(\frac{y - 5}{2}\right)^2 + 1 \). Thus, the inverse function is: \( f^{-1}(y) = \left(\frac{y - 5}{2}\right)^2 + 1 \). Replacing \( y \) back with \( x \) gives: \( f^{-1}(x) = \left(\frac{x - 5}{2}\right)^2 + 1 \). Isn’t it cool how we can reverse functions like this? It’s like flipping a pancake – you get to see the other side! When working with square roots, remember to check for any extraneous solutions. Often, squaring both sides can introduce values that don’t satisfy the original equation, so it’s always good to plug your inverse back into the original function to ensure they match up correctly!