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If [tex] f(x)=\sqrt{\frac{1}{2} x-10+3} [/tex], which inequality can be used to find the domain of [tex] f(x) [/tex]?

A. [tex] \sqrt{\frac{1}{2} x} \geq 0 [/tex]

B. [tex] \frac{1}{2} x \geq 0 [/tex]

C. [tex] \frac{1}{2} x - 10 \geq 0 [/tex]

D. [tex] \sqrt{\frac{1}{2} x - 10} + 3 \geq 0 [/tex]


Sagot :

To find the domain of the function [tex]\( f(x) = \sqrt{\frac{1}{2} x - 10 + 3} \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which the expression inside the square root is non-negative. This is because the square root function is only defined for non-negative inputs.

Let's follow the steps to find the appropriate inequality:

1. Start with the given function:
[tex]\[ f(x) = \sqrt{\frac{1}{2} x - 10 + 3} \][/tex]

2. Simplify the expression inside the square root:
[tex]\[ \frac{1}{2} x - 10 + 3 = \frac{1}{2} x - 7 \][/tex]

3. Set the expression inside the square root to be greater than or equal to zero:
[tex]\[ \frac{1}{2} x - 7 \geq 0 \][/tex]

This inequality ensures that the expression inside the square root is non-negative, which is necessary for the function to be defined.

Therefore, the inequality that can be used to find the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ \frac{1}{2} x - 7 \geq 0 \][/tex]

Thus, the correct option from the given choices is:
[tex]\[ \frac{1}{2} x - 10 \geq 0 \][/tex]
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