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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 focus on the part of the function under the square root, [tex]\(\sqrt{x-7}\)[/tex].

Square roots are only defined for non-negative values, which means the expression inside the square root must be greater than or equal to zero. This gives us the inequality:

[tex]\[ x - 7 \geq 0 \][/tex]

To solve this inequality, we simply add 7 to both sides:

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

This means that for the function [tex]\(h(x)\)[/tex] to be valid, [tex]\(x\)[/tex] must be at least 7 or greater. Therefore, the domain of the function [tex]\(h(x) = \sqrt{x-7} + 5\)[/tex] is all [tex]\(x\)[/tex] such that [tex]\(x \geq 7\)[/tex].

The correct answer is:

D. [tex]\( x \geq 7 \)[/tex]

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