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Answer :
To determine the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to focus on the expression inside the square root, because square roots are only defined for non-negative numbers in the set of real numbers.
1. Identify the constraint on the square root:
- The expression inside the square root is [tex]\( x - 7 \)[/tex].
- For the square root to be defined (and yield a real number), we need:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solve the inequality:
- Add 7 to both sides of the inequality:
[tex]\[
x \geq 7
\][/tex]
3. Determine the domain of the function:
- The domain of [tex]\( h(x) \)[/tex] includes all [tex]\( x \)[/tex] values that satisfy the inequality [tex]\( x \geq 7 \)[/tex].
- This means the domain is all real numbers that are greater than or equal to 7.
4. Choose the correct answer:
- Looking at the provided options:
- A. [tex]\( x \leq 5 \)[/tex]
- B. [tex]\( x \geq 5 \)[/tex]
- C. [tex]\( x \geq 7 \)[/tex]
- D. [tex]\( x \leq -7 \)[/tex]
- The correct choice that matches our solution is option C: [tex]\( x \geq 7 \)[/tex].
Therefore, the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex] is [tex]\( x \geq 7 \)[/tex], which corresponds to option C.
1. Identify the constraint on the square root:
- The expression inside the square root is [tex]\( x - 7 \)[/tex].
- For the square root to be defined (and yield a real number), we need:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solve the inequality:
- Add 7 to both sides of the inequality:
[tex]\[
x \geq 7
\][/tex]
3. Determine the domain of the function:
- The domain of [tex]\( h(x) \)[/tex] includes all [tex]\( x \)[/tex] values that satisfy the inequality [tex]\( x \geq 7 \)[/tex].
- This means the domain is all real numbers that are greater than or equal to 7.
4. Choose the correct answer:
- Looking at the provided options:
- A. [tex]\( x \leq 5 \)[/tex]
- B. [tex]\( x \geq 5 \)[/tex]
- C. [tex]\( x \geq 7 \)[/tex]
- D. [tex]\( x \leq -7 \)[/tex]
- The correct choice that matches our solution is option C: [tex]\( x \geq 7 \)[/tex].
Therefore, the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex] is [tex]\( x \geq 7 \)[/tex], which corresponds to option C.
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