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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 \geq 7[/tex]

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

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

Answer :

To determine the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to find all possible values of [tex]\( x \)[/tex] for which the function is defined.

1. Identify the Restriction:
- The square root function [tex]\(\sqrt{x-7}\)[/tex] is only defined for non-negative values. Therefore, the expression inside the square root, [tex]\(x - 7\)[/tex], must be greater than or equal to zero.

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

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

4. Conclusion:
- The domain of the function [tex]\( h(x) \)[/tex] is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].

Therefore, the correct answer is B. [tex]\( x \geq 7 \)[/tex].

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