Answer :
Sure, let's find the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex].
1. Identify the restriction for the function: The function includes a square root, [tex]\(\sqrt{x - 7}\)[/tex]. The expression inside the square root [tex]\( x - 7 \)[/tex] must be non-negative because the square root of a negative number is not defined in the realm of real numbers.
2. Set up the inequality: To ensure [tex]\( x - 7 \)[/tex] is non-negative,
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality:
[tex]\[
x \geq 7
\][/tex]
4. Interpret the result: 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.
Based on this analysis, the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is:
[tex]\[
\boxed{x \geq 7}
\][/tex]
Therefore, the correct option is:
A. [tex]\(\quad x \geq 7\)[/tex]
1. Identify the restriction for the function: The function includes a square root, [tex]\(\sqrt{x - 7}\)[/tex]. The expression inside the square root [tex]\( x - 7 \)[/tex] must be non-negative because the square root of a negative number is not defined in the realm of real numbers.
2. Set up the inequality: To ensure [tex]\( x - 7 \)[/tex] is non-negative,
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality:
[tex]\[
x \geq 7
\][/tex]
4. Interpret the result: 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.
Based on this analysis, the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is:
[tex]\[
\boxed{x \geq 7}
\][/tex]
Therefore, the correct option is:
A. [tex]\(\quad x \geq 7\)[/tex]
