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]
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]
