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

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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 quadratic function.
   - The function $$ g(x) $$ is **not** one-to-one over all real numbers because it fails the horizontal line test. However, if we restrict the domain to $$ x \geq 0 $$ or $$ x \leq 0 $$, the function becomes one-to-one and thus invertible.

2. **Determine the Range of $$ g(x) $$:**
   - Since $$ x^2 $$ is always non-negative ($$ x^2 \geq 0 $$), adding 5 shifts the graph upward.
   - Therefore, $$ g(x) \geq 5 $$ for all $$ x $$ in the domain.
   - **Range of $$ g(x) $$:** $$[5, \infty)$$

3. **Domain of the Inverse Function:**
   - The domain of the inverse function $$ g^{-1}(y) $$ is the same as the range of the original function $$ g(x) $$.
   - Thus, the domain of $$ g^{-1}(y) $$ is $$[5, \infty)$$.

**Final Answer:**
The inverse function is defined for all real numbers y with y ≥ 5. Thus, its domain is [5 , ∞).
The inverse function $$ g^{-1}(y) $$ is defined for all real numbers $$ y $$ where $$ y \geq 5 $$. Therefore, the domain of the inverse function is $$[5, \infty)$$.

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

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