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

A. [tex] x \leq -7 [/tex]
B. [tex] x \leq 5 [/tex]
C. [tex] x \geq 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 square root part [tex]\(\sqrt{x-7}\)[/tex].

1. Identify domain restrictions: The expression inside the square root needs to be non-negative, meaning it cannot be less than zero. So, we set up the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]

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

3. Conclusion: The domain of the function [tex]\( h(x) \)[/tex] is all [tex]\( x \)[/tex] values that satisfy [tex]\( x \geq 7 \)[/tex].

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

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