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, because we know that we can't take the square root of a negative number in the set of real numbers.
Here are the steps:
1. Look at the expression inside the square root: [tex]\( x - 7 \)[/tex].
2. Set up an inequality to find when this expression is non-negative (greater than or equal to zero):
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality:
- Add 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
Since [tex]\( x \)[/tex] must be greater than or equal to 7 for the square root to be real, the domain of the function [tex]\( h(x) \)[/tex] is all [tex]\( x \)[/tex] values such that [tex]\( x \geq 7 \)[/tex].
Therefore, the correct answer is:
C. [tex]\( x \geq 7 \)[/tex]
Here are the steps:
1. Look at the expression inside the square root: [tex]\( x - 7 \)[/tex].
2. Set up an inequality to find when this expression is non-negative (greater than or equal to zero):
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality:
- Add 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
Since [tex]\( x \)[/tex] must be greater than or equal to 7 for the square root to be real, the domain of the function [tex]\( h(x) \)[/tex] is all [tex]\( x \)[/tex] values such that [tex]\( x \geq 7 \)[/tex].
Therefore, the correct answer is:
C. [tex]\( x \geq 7 \)[/tex]
