QUESTION 8 - 1 POINT The function \[ f(x)=-2 \sqrt{x+3}+8 \quad x \geq-3 \] has an inverse \( f^{-1}(x) \) defined on the domain \( x \leq 8 \). Find the inverse. Provide

To find the inverse, we start by replacing \( f(x) \) with \( y \): \[ y = -2 \sqrt{x+3} + 8. \] Now, solve for \( x \) in terms of \( y \): 1. Subtract 8 from both sides: \[ y - 8 = -2 \sqrt{x+3}. \] 2. Divide by -2: \[ \sqrt{x+3} = \frac{8 - y}{2}. \] 3. Square both sides: \[ x + 3 = \left(\frac{8 - y}{2}\right)^2. \] 4. Subtract 3 from both sides: \[ x = \left(\frac{8 - y}{2}\right)^2 - 3. \] Now, let’s express the inverse function: \[ f^{-1}(y) = \left(\frac{8 - y}{2}\right)^2 - 3. \] Replace \( y \) by \( x \) to express the inverse function in conventional notation: \[ f^{-1}(x) = \left(\frac{8 - x}{2}\right)^2 - 3. \] And there you have it! The inverse function is: \[ f^{-1}(x) = \left(\frac{8 - x}{2}\right)^2 - 3 \quad \text{for } x \leq 8. \]