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Answer :
To find the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to ensure that all the expressions inside the square root are defined and real numbers.
The square root function, [tex]\(\sqrt{y}\)[/tex], is only defined for non-negative values of [tex]\( y \)[/tex]. This means that the expression inside the square root, [tex]\( x - 7 \)[/tex], must be non-negative. Therefore, we must solve the inequality:
[tex]\[ x - 7 \geq 0 \][/tex]
To solve for [tex]\( x \)[/tex], we add 7 to both sides of the inequality:
[tex]\[ x \geq 7 \][/tex]
Thus, the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex] includes all [tex]\( x \)[/tex] values that are greater than or equal to 7. In set notation, this is expressed as [tex]\( \{ x \mid x \geq 7 \} \)[/tex].
Now, let's look at the given options:
A. [tex]\( x \geq 7 \)[/tex]
B. [tex]\( x \leq 5 \)[/tex]
C. [tex]\( x \geq 5 \)[/tex]
D. [tex]\( x \leq -7 \)[/tex]
The correct choice is:
A. [tex]\( x \geq 7 \)[/tex]
So, the domain of the function [tex]\( h \)[/tex] is [tex]\( x \geq 7 \)[/tex].
The square root function, [tex]\(\sqrt{y}\)[/tex], is only defined for non-negative values of [tex]\( y \)[/tex]. This means that the expression inside the square root, [tex]\( x - 7 \)[/tex], must be non-negative. Therefore, we must solve the inequality:
[tex]\[ x - 7 \geq 0 \][/tex]
To solve for [tex]\( x \)[/tex], we add 7 to both sides of the inequality:
[tex]\[ x \geq 7 \][/tex]
Thus, the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex] includes all [tex]\( x \)[/tex] values that are greater than or equal to 7. In set notation, this is expressed as [tex]\( \{ x \mid x \geq 7 \} \)[/tex].
Now, let's look at the given options:
A. [tex]\( x \geq 7 \)[/tex]
B. [tex]\( x \leq 5 \)[/tex]
C. [tex]\( x \geq 5 \)[/tex]
D. [tex]\( x \leq -7 \)[/tex]
The correct choice is:
A. [tex]\( x \geq 7 \)[/tex]
So, the domain of the function [tex]\( h \)[/tex] is [tex]\( x \geq 7 \)[/tex].
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