High School

Select the correct answer.

What is the domain of the function [tex]h[/tex]?

[tex]h(x) = \sqrt{x-7} + 5[/tex]

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

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

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

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

Answer :

To determine the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to analyze the expression under the square root. The square root function is only defined for non-negative numbers, which means the expression inside the square root must be greater than or equal to zero.

The expression under the square root is [tex]\( x - 7 \)[/tex]. We set up the inequality:

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

Now, we solve this inequality to find the permissible values for [tex]\( x \)[/tex]:

1. Add 7 to both sides of the inequality:

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

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

This tells us that [tex]\( x \)[/tex] must be greater than or equal to 7 for [tex]\( h(x) \)[/tex] to be defined. Hence, the domain of the function is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].

Therefore, the correct answer is:

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