High School

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 consider when the expression inside the square root is non-negative. This is because the square root of a negative number is not defined in the set of real numbers.

Here are the steps to find the domain:

1. Identify the expression inside the square root: For the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], the expression inside the square root is [tex]\( x - 7 \)[/tex].

2. Set the expression greater than or equal to zero: We need [tex]\( x - 7 \geq 0 \)[/tex]. This ensures that the expression inside the square root is non-negative, which is necessary for the function to be defined.

3. Solve the inequality:
- [tex]\( x - 7 \geq 0 \)[/tex]
- Add 7 to both sides: [tex]\( x \geq 7 \)[/tex]

4. State the domain based on the inequality: The solution [tex]\( x \geq 7 \)[/tex] describes the domain of the function. This means all [tex]\( x \)[/tex]-values greater than or equal to 7 will make the function [tex]\( h(x) \)[/tex] defined.

Thus, the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is [tex]\( x \geq 7 \)[/tex]. The correct answer, corresponding to this domain, is:

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