Answer :
To find the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to ensure that the expression inside the square root is non-negative. This is because we cannot take the square root of negative numbers in real number arithmetic.
Here are the steps to determine the domain:
1. Expression inside the square root: We have to evaluate the expression [tex]\( x - 7 \)[/tex] which is inside the square root.
2. Set up the inequality: For the square root to be valid, [tex]\( x - 7 \)[/tex] must be greater than or equal to zero. Thus, we write the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality: To solve for [tex]\( x \)[/tex], we add 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
Therefore, the domain of the function [tex]\( h(x) \)[/tex] is all [tex]\( x \)[/tex] values such that [tex]\( x \geq 7 \)[/tex].
This corresponds to option A: [tex]\( x \geq 7 \)[/tex]. So, the domain of the function is [tex]\( x \geq 7 \)[/tex].
Here are the steps to determine the domain:
1. Expression inside the square root: We have to evaluate the expression [tex]\( x - 7 \)[/tex] which is inside the square root.
2. Set up the inequality: For the square root to be valid, [tex]\( x - 7 \)[/tex] must be greater than or equal to zero. Thus, we write the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality: To solve for [tex]\( x \)[/tex], we add 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
Therefore, the domain of the function [tex]\( h(x) \)[/tex] is all [tex]\( x \)[/tex] values such that [tex]\( x \geq 7 \)[/tex].
This corresponds to option A: [tex]\( x \geq 7 \)[/tex]. So, the domain of the function is [tex]\( x \geq 7 \)[/tex].
