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Answer :
To determine the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], follow these steps:
1. Identify the condition for the square root: The expression inside the square root, [tex]\( x-7 \)[/tex], must be non-negative. This is because the square root of a negative number is not a real number.
2. Set up the inequality: To ensure [tex]\( x-7 \)[/tex] is non-negative, we need:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality: Add 7 to both sides of the inequality:
[tex]\[
x \geq 7
\][/tex]
4. Conclusion: Therefore, the domain of the function [tex]\( h(x) \)[/tex] is all [tex]\( x \)[/tex] values that are greater than or equal to 7.
Given the provided options:
- A. [tex]\( x \geq 5 \)[/tex]
- B. [tex]\( x \geq 7 \)[/tex]
- C. [tex]\( x \leq -7 \)[/tex]
- D. [tex]\( x \leq 5 \)[/tex]
The correct answer is:
B. [tex]\( x \geq 7 \)[/tex]
1. Identify the condition for the square root: The expression inside the square root, [tex]\( x-7 \)[/tex], must be non-negative. This is because the square root of a negative number is not a real number.
2. Set up the inequality: To ensure [tex]\( x-7 \)[/tex] is non-negative, we need:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality: Add 7 to both sides of the inequality:
[tex]\[
x \geq 7
\][/tex]
4. Conclusion: Therefore, the domain of the function [tex]\( h(x) \)[/tex] is all [tex]\( x \)[/tex] values that are greater than or equal to 7.
Given the provided options:
- A. [tex]\( x \geq 5 \)[/tex]
- B. [tex]\( x \geq 7 \)[/tex]
- C. [tex]\( x \leq -7 \)[/tex]
- D. [tex]\( x \leq 5 \)[/tex]
The correct answer is:
B. [tex]\( x \geq 7 \)[/tex]
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