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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 \leq 5[/tex]
D. [tex]x \geq 7[/tex]

Answer :

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

Here's how you can find the domain step-by-step:

1. Identify the Expression Under the Square Root:
The expression inside the square root is [tex]\( x - 7 \)[/tex].

2. Set Up the Inequality:
For the square root to be defined in the real numbers, the expression under the square root must be greater than or equal to zero:
[tex]\[
x - 7 \geq 0
\][/tex]

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

4. Determine the Domain:
The function [tex]\( h(x) \)[/tex] is defined for all [tex]\( x \)[/tex] values that satisfy the inequality.
Therefore, the domain of [tex]\( h(x) \)[/tex] is:
[tex]\[
x \geq 7
\][/tex]

So, the correct answer is:
- D. [tex]\( x \geq 7 \)[/tex]

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