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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 \geq 5[/tex]

B. [tex]x \leq 5[/tex]

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

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

Answer :

Let's determine the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex].

For the square root function [tex]\( \sqrt{a} \)[/tex] to be real and defined, the expression inside the square root, [tex]\( a \)[/tex], must be non-negative ([tex]\( a \geq 0 \)[/tex]).

In our function, the expression inside the square root is [tex]\( x - 7 \)[/tex]. Therefore, we set up the following inequality:

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

Now, let's solve for [tex]\( x \)[/tex]:

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

This means that the domain of the function [tex]\( h(x) \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].

Therefore, the correct answer is:
[tex]\[ \boxed{D. \quad x \geq 7} \][/tex]

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