Answer :
To determine the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], we need to consider the expression inside the square root. The square root function is only defined for non-negative numbers, meaning the expression inside the square root must be greater than or equal to zero.
Here's a step-by-step breakdown:
1. Identify the expression inside the square root: In the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], the expression inside the square root is [tex]\( x - 7 \)[/tex].
2. Set up an inequality: Since the expression inside the square root must be non-negative, we have:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality: To find the values of [tex]\( x \)[/tex] that satisfy this inequality, solve for [tex]\( x \)[/tex]:
[tex]\[
x - 7 \geq 0 \\
x \geq 7
\][/tex]
4. Determine the domain: The solution to this inequality tells us that the domain of the function [tex]\( h(x) \)[/tex] is [tex]\( x \geq 7 \)[/tex].
Therefore, the correct answer for the domain of the given function is:
C. [tex]\( x \geq 7 \)[/tex]
Here's a step-by-step breakdown:
1. Identify the expression inside the square root: In the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], the expression inside the square root is [tex]\( x - 7 \)[/tex].
2. Set up an inequality: Since the expression inside the square root must be non-negative, we have:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality: To find the values of [tex]\( x \)[/tex] that satisfy this inequality, solve for [tex]\( x \)[/tex]:
[tex]\[
x - 7 \geq 0 \\
x \geq 7
\][/tex]
4. Determine the domain: The solution to this inequality tells us that the domain of the function [tex]\( h(x) \)[/tex] is [tex]\( x \geq 7 \)[/tex].
Therefore, the correct answer for the domain of the given function is:
C. [tex]\( x \geq 7 \)[/tex]
