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------------------------------------------------ Select the correct answer.

What is the domain of the function [tex]h[/tex]?

[tex]h(x) = \sqrt{x-7} + 5[/tex]

A. [tex]x \leq 5[/tex]

B. [tex]x \leq -7[/tex]

C. [tex]x \geq 7[/tex]

D. [tex]x \geq 5[/tex]

To determine the domain of the function [tex]$ h(x) = \sqrt{x - 7} + 5 $[/tex], we need to consider the expression inside the square root. The square root function is only defined for non-negative numbers, meaning the expression inside the square root must be greater than or equal to zero.

Here's a step-by-step breakdown:

1. Identify the expression inside the square root: In the function [tex]$ h(x) = \sqrt{x - 7} + 5 $[/tex], the expression inside the square root is [tex]$ x - 7 $[/tex].

2. Set up an inequality: Since the expression inside the square root must be non-negative, we have:
[tex]\[
x - 7 \geq 0
\][/tex]

3. Solve the inequality: To find the values of [tex]$ x $[/tex] that satisfy this inequality, solve for [tex]$ x $[/tex]:
[tex]\[
x - 7 \geq 0 \\
x \geq 7
\][/tex]

4. Determine the domain: The solution to this inequality tells us that the domain of the function [tex]$ h(x) $[/tex] is [tex]$ x \geq 7 $[/tex].

Therefore, the correct answer for the domain of the given function is:

C. [tex]$ x \geq 7 $[/tex]