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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]

Answer :

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

Here’s how you can find the domain:

1. Set up the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]

2. Solve for [tex]\( x \)[/tex]:
Add 7 to both sides of the inequality:
[tex]\[
x \geq 7
\][/tex]

3. Interpret the solution:
This tells us that [tex]\( x \)[/tex] must be greater than or equal to 7 for the function to be defined.

Thus, the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex] is [tex]\( x \geq 7 \)[/tex].

The correct answer is:
C. [tex]\( x \geq 7 \)[/tex]

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