Answer :
To determine the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], we need to look at where the function is defined.
1. Square Root Function Condition:
The square root function, [tex]\( \sqrt{\cdot} \)[/tex], is defined only for non-negative values. For [tex]\( \sqrt{x - 7} \)[/tex] to be valid, the expression inside the square root must be non-negative:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solving the Inequality:
Solve the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
Adding 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
Therefore, the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is defined for all [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
Thus, the domain of the function [tex]\( h \)[/tex] is:
[tex]\[
\boxed{x \geq 7}
\][/tex]
From the given choices, the correct one is:
C. [tex]\( x \geq 7 \)[/tex]
1. Square Root Function Condition:
The square root function, [tex]\( \sqrt{\cdot} \)[/tex], is defined only for non-negative values. For [tex]\( \sqrt{x - 7} \)[/tex] to be valid, the expression inside the square root must be non-negative:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solving the Inequality:
Solve the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
Adding 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
Therefore, the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is defined for all [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
Thus, the domain of the function [tex]\( h \)[/tex] is:
[tex]\[
\boxed{x \geq 7}
\][/tex]
From the given choices, the correct one is:
C. [tex]\( x \geq 7 \)[/tex]
