Answer :
To find the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to ensure that any expression under the square root is non-negative, because square root is only defined for non-negative numbers in the set of real numbers.
Here's how you determine it step-by-step:
1. Look at the expression inside the square root: [tex]\( x - 7 \)[/tex].
2. Set up the inequality so that expression under the square root is greater than or equal to zero:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality for [tex]\( x \)[/tex]:
[tex]\[
x \geq 7
\][/tex]
Therefore, the domain of the function [tex]\( h \)[/tex] consists of all values of [tex]\( x \)[/tex] that are greater than or equal to 7. In interval notation, this is written as [tex]\([7, \infty)\)[/tex].
From the given options, the correct choice is:
- D. [tex]\(x \geq 7\)[/tex]
This means the function is defined for all real numbers [tex]\( x \)[/tex] that are 7 or larger, ensuring that the square root operation is valid.
Here's how you determine it step-by-step:
1. Look at the expression inside the square root: [tex]\( x - 7 \)[/tex].
2. Set up the inequality so that expression under the square root is greater than or equal to zero:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality for [tex]\( x \)[/tex]:
[tex]\[
x \geq 7
\][/tex]
Therefore, the domain of the function [tex]\( h \)[/tex] consists of all values of [tex]\( x \)[/tex] that are greater than or equal to 7. In interval notation, this is written as [tex]\([7, \infty)\)[/tex].
From the given options, the correct choice is:
- D. [tex]\(x \geq 7\)[/tex]
This means the function is defined for all real numbers [tex]\( x \)[/tex] that are 7 or larger, ensuring that the square root operation is valid.
