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 important because the square root is only defined for non-negative numbers in real numbers.
Here's the step-by-step process:
1. Identify the condition inside the square root:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
To make sure the expression inside the square root is non-negative, solve the inequality:
[tex]\[
x \geq 7
\][/tex]
3. Determine the domain:
This means that the function [tex]\( h(x) \)[/tex] is defined for all [tex]\( x \)[/tex] values that are greater than or equal to 7. Therefore, the domain of [tex]\( h(x) \)[/tex] is [tex]\( x \geq 7 \)[/tex].
Now, based on the options provided:
A. [tex]\( x \leq 5 \)[/tex]
B. [tex]\( x \geq 7 \)[/tex]
C. [tex]\( x \geq 5 \)[/tex]
D. [tex]\( z \leq -7 \)[/tex]
The correct answer is B. [tex]\( x \geq 7 \)[/tex]. This option matches the condition for the domain of the function.
Here's the step-by-step process:
1. Identify the condition inside the square root:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
To make sure the expression inside the square root is non-negative, solve the inequality:
[tex]\[
x \geq 7
\][/tex]
3. Determine the domain:
This means that the function [tex]\( h(x) \)[/tex] is defined for all [tex]\( x \)[/tex] values that are greater than or equal to 7. Therefore, the domain of [tex]\( h(x) \)[/tex] is [tex]\( x \geq 7 \)[/tex].
Now, based on the options provided:
A. [tex]\( x \leq 5 \)[/tex]
B. [tex]\( x \geq 7 \)[/tex]
C. [tex]\( x \geq 5 \)[/tex]
D. [tex]\( z \leq -7 \)[/tex]
The correct answer is B. [tex]\( x \geq 7 \)[/tex]. This option matches the condition for the domain of the function.
