If the function \( g(x) = x^{2} + 5 \) has an inverse, determine the domain of the inverse function.

To determine the domain of the inverse function of \( g(x) = x^{2} + 5 \), let's follow these steps: 1. **Understand the Original Function:** - \( g(x) = x^{2} + 5 \) is a parabola that opens upward with its vertex at \( (0, 5) \). - **Domain of \( g(x) \):** All real numbers (\( \mathbb{R} \)). - **Range of \( g(x) \):** Since the parabola opens upward, the minimum value is \( 5 \). Thus, the range is \( [5, \infty) \). 2. **Invertibility:** - For \( g(x) \) to have an inverse, it must be **one-to-one** (bijective). - The original function \( g(x) = x^{2} + 5 \) is **not** one-to-one over all real numbers because it fails the horizontal line test. - To make it invertible, we typically **restrict the domain**. For example: - If we restrict \( g(x) \) to \( x \geq 0 \), the function becomes one-to-one. - Similarly, restricting to \( x \leq 0 \) also makes it one-to-one. - Regardless of the restriction, the **range remains** \( [5, \infty) \). 3. **Domain of the Inverse Function:** - The domain of the inverse function \( g^{-1}(y) \) is the **range of the original function \( g(x) \)**. - Therefore, the domain of \( g^{-1}(y) \) is \( [5, \infty) \). **Final Answer:** All real numbers y with y at least five. In other words, the domain is [ 5,  ∞ ).