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What is the domain of the function [tex]h(x) = \sqrt{x-7} + 5[/tex]?

A. [tex]x \geq 5[/tex]
B. [tex]x \leq -7[/tex]
C. [tex]x \geq 7[/tex]
D. [tex]x \leq 5[/tex]

Answer :

To find the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], we need to make sure that the expression inside the square root, [tex]\( x - 7 \)[/tex], is non-negative. This is because the square root of a negative number is not defined in the set of real numbers.

Here are the steps to determine the domain:

1. Set Up the Inequality:
[tex]\[ x - 7 \geq 0 \][/tex]

2. Solve for [tex]\( x \)[/tex]:
To solve this inequality, add 7 to both sides:
[tex]\[ x \geq 7 \][/tex]

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

Therefore, the domain of the function [tex]\( h(x) \)[/tex] is [tex]\( x \geq 7 \)[/tex].

Looking at the options provided:

- A. [tex]\( x \geq 5 \)[/tex]
- B. [tex]\( x \leq -7 \)[/tex]
- C. [tex]\( x \geq 7 \)[/tex]
- D. [tex]\( x \leq 5 \)[/tex]

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

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