To determine the range of the function [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex], let's break down the various components of the function and analyze their effects.
1. Understanding the Square Root Function:
- The square root function, [tex]\( \sqrt{x} \)[/tex], is only defined for non-negative values of [tex]\( x \)[/tex]. This means that [tex]\( x \)[/tex] must be greater than or equal to 0.
- The output of [tex]\( \sqrt{x} \)[/tex] is also non-negative. This means that [tex]\( \sqrt{x} \)[/tex] ranges from 0 to [tex]\( \infty \)[/tex].
2. Applying the Coefficient of [tex]\(\frac{1}{2}\)[/tex]:
- When we multiply the output of [tex]\( \sqrt{x} \)[/tex] by [tex]\(\frac{1}{2}\)[/tex], we are essentially scaling the outputs of the square root function by [tex]\( \frac{1}{2} \)[/tex].
- The smallest value [tex]\( \sqrt{x} \)[/tex] can take is 0, and [tex]\( \frac{1}{2} \cdot 0 = 0 \)[/tex].
- The output value of [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] will start at 0 (when [tex]\( x = 0 \)[/tex]) and can increase without bound, approaching [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] increases.
3. Conclusion:
- The function [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] produces values starting from 0 and increasing to infinity.
- Therefore, the range of the function [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] is [tex]\( \{ y \in \mathbb{R} \, | \, y \geq 0 \} \)[/tex], which translates to all real numbers greater than or equal to 0.
Given this analysis, the range of the function [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] is:
All real numbers greater than or equal to 0
