Answer :
To determine the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], we need to identify the values of [tex]\( x \)[/tex] for which the function is defined.
1. Understanding the Square Root Function:
The expression inside the square root, [tex]\( x - 7 \)[/tex], must be greater than or equal to zero because the square root of a negative number is not defined in the real number system.
2. Setting Up the Inequality:
We solve the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solving the Inequality:
Add 7 to both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x \geq 7
\][/tex]
4. Conclusion:
The domain of the function is the set of all real numbers [tex]\( x \)[/tex] that satisfy the inequality [tex]\( x \geq 7 \)[/tex].
Therefore, the correct answer is:
- A. [tex]\( x \geq 7 \)[/tex]
This means the function [tex]\( h(x) \)[/tex] is defined for all [tex]\( x \)[/tex] starting from 7 and extending to positive infinity.
1. Understanding the Square Root Function:
The expression inside the square root, [tex]\( x - 7 \)[/tex], must be greater than or equal to zero because the square root of a negative number is not defined in the real number system.
2. Setting Up the Inequality:
We solve the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solving the Inequality:
Add 7 to both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x \geq 7
\][/tex]
4. Conclusion:
The domain of the function is the set of all real numbers [tex]\( x \)[/tex] that satisfy the inequality [tex]\( x \geq 7 \)[/tex].
Therefore, the correct answer is:
- A. [tex]\( x \geq 7 \)[/tex]
This means the function [tex]\( h(x) \)[/tex] is defined for all [tex]\( x \)[/tex] starting from 7 and extending to positive infinity.
