College

Select the correct answer.

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

A. [tex]x \leq -7[/tex]
B. [tex]x \geq 5[/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 ensure that the expression inside the square root, [tex]\( x-7 \)[/tex], 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 we find the domain:

1. Set Up the Inequality:
We want the expression inside the square root to be greater than or equal to zero:
[tex]\[
x - 7 \geq 0
\][/tex]

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

3. Determine the Domain:
The solution to the inequality [tex]\( x \geq 7 \)[/tex] means that [tex]\( x \)[/tex] can be any number greater than or equal to 7. This ensures the expression inside the square root is non-negative.

Therefore, the domain of the function is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].

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