To determine the range of the function [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex], let's analyze it step by step:
1. Understanding the function:
The function [tex]\( f(x) \)[/tex] involves the square root of [tex]\( x \)[/tex]. The square root function, [tex]\(\sqrt{x}\)[/tex], is only defined for [tex]\( x \geq 0 \)[/tex] because you cannot take the square root of a negative number within the real number system.
2. Behavior for different values of [tex]\( x \)[/tex]:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = \frac{1}{2} \sqrt{0} = \frac{1}{2} \cdot 0 = 0
\][/tex]
- When [tex]\( x > 0 \)[/tex]:
The value of [tex]\(\sqrt{x}\)[/tex] is positive and thus the value of [tex]\( \frac{1}{2} \sqrt{x} \)[/tex] will also be positive but equal to half of the square root of [tex]\( x \)[/tex].
3. Formulating the range:
- As [tex]\( x \)[/tex] increases from 0 to positive infinity, the value of [tex]\( \sqrt{x} \)[/tex] also increases from 0 to positive infinity.
- Consequently, [tex]\( \frac{1}{2} \sqrt{x} \)[/tex] also increases from 0 to positive infinity.
Therefore, since [tex]\( f(x) \)[/tex] starts at 0 (when [tex]\( x = 0 \)[/tex]) and increases without bound as [tex]\( x \)[/tex] increases, the range of the function [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] is all real numbers greater than or equal to 0.
In conclusion, the range of the function [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] is:
[tex]\[ \text{all real numbers greater than or equal to 0} \][/tex]
