Answer :
To find the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], we need to make sure that the expression inside the square root, [tex]\( x - 7 \)[/tex], is non-negative. This is because the square root of a negative number is not defined in the set of real numbers.
Here are the steps to determine the domain:
1. Set Up the Inequality:
[tex]\[ x - 7 \geq 0 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
To solve this inequality, add 7 to both sides:
[tex]\[ x \geq 7 \][/tex]
3. Conclusion:
This tells us that the values of [tex]\( x \)[/tex] must be greater than or equal to 7 for the function [tex]\( h(x) \)[/tex] to be defined.
Therefore, the domain of the function [tex]\( h(x) \)[/tex] is [tex]\( x \geq 7 \)[/tex].
Looking at the options provided:
- A. [tex]\( x \geq 5 \)[/tex]
- B. [tex]\( x \leq -7 \)[/tex]
- C. [tex]\( x \geq 7 \)[/tex]
- D. [tex]\( x \leq 5 \)[/tex]
The correct answer is:
C. [tex]\( x \geq 7 \)[/tex]
Here are the steps to determine the domain:
1. Set Up the Inequality:
[tex]\[ x - 7 \geq 0 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
To solve this inequality, add 7 to both sides:
[tex]\[ x \geq 7 \][/tex]
3. Conclusion:
This tells us that the values of [tex]\( x \)[/tex] must be greater than or equal to 7 for the function [tex]\( h(x) \)[/tex] to be defined.
Therefore, the domain of the function [tex]\( h(x) \)[/tex] is [tex]\( x \geq 7 \)[/tex].
Looking at the options provided:
- A. [tex]\( x \geq 5 \)[/tex]
- B. [tex]\( x \leq -7 \)[/tex]
- C. [tex]\( x \geq 7 \)[/tex]
- D. [tex]\( x \leq 5 \)[/tex]
The correct answer is:
C. [tex]\( x \geq 7 \)[/tex]
