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Answer :
To determine the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], we need to consider when the expression under the square root is non-negative. This is because the square root of a negative number is not defined in the set of real numbers.
1. Start by identifying the expression inside the square root: [tex]\( x - 7 \)[/tex].
2. Set up the inequality for when this expression is non-negative:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality:
[tex]\[
x \geq 7
\][/tex]
This tells us that the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is defined for all values of [tex]\( x \)[/tex] that are greater than or equal to 7.
Therefore, the domain of the function is [tex]\( x \geq 7 \)[/tex].
By looking at the given options:
- A. [tex]\( x \geq 5 \)[/tex]
- B. [tex]\( x \leq 5 \)[/tex]
- C. [tex]\( x \leq -7 \)[/tex]
- D. [tex]\( x \geq 7 \)[/tex]
The correct answer is:
- D. [tex]\( x \geq 7 \)[/tex]
1. Start by identifying the expression inside the square root: [tex]\( x - 7 \)[/tex].
2. Set up the inequality for when this expression is non-negative:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality:
[tex]\[
x \geq 7
\][/tex]
This tells us that the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is defined for all values of [tex]\( x \)[/tex] that are greater than or equal to 7.
Therefore, the domain of the function is [tex]\( x \geq 7 \)[/tex].
By looking at the given options:
- A. [tex]\( x \geq 5 \)[/tex]
- B. [tex]\( x \leq 5 \)[/tex]
- C. [tex]\( x \leq -7 \)[/tex]
- D. [tex]\( x \geq 7 \)[/tex]
The correct answer is:
- D. [tex]\( x \geq 7 \)[/tex]
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