What is the range of the inverse function for the function $$ f(x) = \sqrt{x - 4} + 1 $$?

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To determine the range of the inverse function for $$ f(x) = \sqrt{x - 4} + 1 $$, follow these steps:

1. **Find the Domain of $$ f(x) $$:**
   - The expression inside the square root must be non-negative:
     $$
     x - 4 \geq 0 \implies x \geq 4
     $$
   - So, the domain of $$ f(x) $$ is $$ [4, \infty) $$.

2. **Determine the Range of $$ f(x) $$:**
   - As $$ x $$ increases from 4 to $$ \infty $$, $$ \sqrt{x - 4} $$ increases from 0 to $$ \infty $$.
   - Therefore, $$ f(x) = \sqrt{x - 4} + 1 $$ increases from $$ 1 $$ to $$ \infty $$.
   - The range of $$ f(x) $$ is $$ [1, \infty) $$.

3. **Find the Inverse Function $$ f^{-1}(y) $$:**
   - Start with $$ y = \sqrt{x - 4} + 1 $$.
   - Solve for $$ x $$:
     $$
     y - 1 = \sqrt{x - 4} \
     (y - 1)^2 = x - 4 \
     x = (y - 1)^2 + 4
     $$
   - So, the inverse function is $$ f^{-1}(y) = (y - 1)^2 + 4 $$.

4. **Determine the Range of the Inverse Function:**
   - The range of the inverse function $$ f^{-1}(y) $$ is the domain of the original function $$ f(x) $$, which is $$ [4, \infty) $$.

**Final Answer:**  
All real numbers y with y at least four. In other words, the range is [ 4, ∞ ).
The range of the inverse function is all real numbers $$ y $$ where $$ y \geq 4 $$.

What is the range of the inverse function for the function \( f(x) = \sqrt{x - 4} + 1 \)?

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