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 5[/tex]
B. [tex]x \geq 7[/tex]
C. [tex]x \leq -7[/tex]
D. [tex]x \geq 5[/tex]

Answer :

To determine the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to focus on the expression inside the square root, [tex]\(\sqrt{x-7}\)[/tex].

Here's a step-by-step guide to finding the domain:

1. Understand the Square Root Requirement:
- For a square root function to be defined, the expression inside the square root must be non-negative (i.e., it cannot be negative).
- Therefore, we need the expression [tex]\( x - 7 \)[/tex] to be greater than or equal to zero.

2. Set Up the Inequality:
- We form the inequality: [tex]\( x - 7 \geq 0 \)[/tex].

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

4. Determine the Domain:
- The inequality [tex]\( x \geq 7 \)[/tex] tells us the values of [tex]\( x \)[/tex] for which the function [tex]\( h(x) \)[/tex] is defined.
- This means the function can take any value of [tex]\( x \)[/tex] as long as [tex]\( x \)[/tex] is 7 or greater.

5. Conclude the Solution:
- Therefore, the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex] is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].

Accordingly, the correct answer is:
B. [tex]\( x \geq 7 \)[/tex]