College

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 find the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which this function is defined.

1. Look at the square root: The expression inside the square root, [tex]\( \sqrt{x - 7} \)[/tex], must be non-negative because square roots of negative numbers are not defined in the set of real numbers.

2. Set up the inequality: To ensure that the expression [tex]\( x - 7 \)[/tex] is non-negative, we set up the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]

3. Solve the inequality: Solve for [tex]\( x \)[/tex] by adding 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 that the values of [tex]\( x \)[/tex] that make the function [tex]\( h(x) \)[/tex] defined are 7 and any number greater than 7.

Thus, the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is greater than or equal to 7. Therefore, the correct answer is:

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