What is the domain of the function [tex]f(x)=\frac{2}{5} \sqrt{x}[/tex]?

A. all real numbers

B. all real numbers less than 0

C. all real numbers less than or equal to 0

D. all real numbers greater than or equal to 0



Answer :

To determine the domain of the function [tex]\( f(x) = \frac{2}{5} \sqrt{x} \)[/tex], we need to consider the properties of the components of the function. The key part to focus on here is the square root function, [tex]\(\sqrt{x}\)[/tex].

1. Square Root Function Analysis: The square root function, [tex]\(\sqrt{x}\)[/tex], is only defined for non-negative values of [tex]\(x\)[/tex]. This means [tex]\(x\)[/tex] must be greater than or equal to 0 for [tex]\(\sqrt{x}\)[/tex] to be real and defined.

2. Function Definition: Since [tex]\(\frac{2}{5}\)[/tex] is just a constant multiplier, it does not affect the domain of [tex]\(\sqrt{x}\)[/tex]. Therefore, the function [tex]\( f(x) = \frac{2}{5} \sqrt{x} \)[/tex] is defined whenever [tex]\(\sqrt{x}\)[/tex] is defined, i.e., for all [tex]\(x\)[/tex] such that [tex]\(x \geq 0\)[/tex].

3. Conclusion: Combining these observations, we see that the function [tex]\( f(x) = \frac{2}{5} \sqrt{x} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex] that are greater than or equal to 0.

Therefore, the domain of the function [tex]\( f(x) = \frac{2}{5} \sqrt{x} \)[/tex] is:

[tex]\(\boxed{\text{all real numbers greater than or equal to 0}}\)[/tex]