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 \( g(x) = x^{2} + 5 \), we first note that \( g(x) \) is a quadratic function. However, it is not one-to-one over its entire domain (which is all real numbers), so we must restrict the domain to make it invertible. Typically, we restrict \( g(x) \) to \( x \geq 0 \) (the right half of the parabola), making the function \( g(x) \) strictly increasing. The minimum value of \( g(x) \) occurs at \( x = 0 \), giving \( g(0) = 5 \). Therefore, the range of \( g(x) \), which will correspond to the domain of the inverse function, is \( [5, \infty) \). So, the domain of the inverse function \( g^{-1}(y) \) is \( [5, \infty) \).