\( \left. \begin{array} { l } { f ( x ) = ( 3 + x ) ^ { 2 } + 5 } \\ { f ^ { - 1 } = \ldots } \end{array} \right. \)

To find the inverse function \( f^{-1}(x) \) for the function \( f(x) = (3 + x)^2 + 5 \), we will follow these steps: 1. **Set \( y = f(x) \)**: \[ y = (3 + x)^2 + 5 \] 2. **Solve for \( x \) in terms of \( y \)**: - First, subtract 5 from both sides: \[ y - 5 = (3 + x)^2 \] - Next, take the square root of both sides. Remember to consider both the positive and negative roots: \[ \sqrt{y - 5} = 3 + x \quad \text{or} \quad -\sqrt{y - 5} = 3 + x \] - Solve for \( x \): \[ x = \sqrt{y - 5} - 3 \quad \text{or} \quad x = -\sqrt{y - 5} - 3 \] 3. **Determine the correct branch**: Since \( f(x) = (3 + x)^2 + 5 \) is a parabola that opens upwards and has its vertex at \( x = -3 \), it is increasing for \( x \geq -3 \). Therefore, we will only consider the positive root: \[ x = \sqrt{y - 5} - 3 \] 4. **Express \( f^{-1}(x) \)**: Now, we can express the inverse function: \[ f^{-1}(x) = \sqrt{x - 5} - 3 \] Thus, the inverse function is: \[ f^{-1}(x) = \sqrt{x - 5} - 3 \]