To determine the range of the function [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex], we need to understand how the function behaves for all possible inputs [tex]\( x \)[/tex].
First, let's analyze the function step-by-step:
1. Understanding the square root function [tex]\(\sqrt{x}\)[/tex]:
- The domain of [tex]\(\sqrt{x}\)[/tex] is [tex]\( x \geq 0 \)[/tex]. This is because the square root function is only defined for non-negative numbers.
- The square root function [tex]\(\sqrt{x}\)[/tex] will produce values starting from 0 and increasing as [tex]\( x \)[/tex] increases. Therefore, the range of [tex]\(\sqrt{x}\)[/tex] is all real numbers greater than or equal to 0, i.e., [tex]\([0, \infty)\)[/tex].
2. Multiplying by [tex]\(\frac{1}{2}\)[/tex]:
- Now consider the function [tex]\( \frac{1}{2} \sqrt{x} \)[/tex].
- Multiplying [tex]\(\sqrt{x}\)[/tex] by [tex]\(\frac{1}{2}\)[/tex] scales the output but does not change the fact that the output starts from 0 and increases as [tex]\( x \)[/tex] increases.
- Therefore, [tex]\(\frac{1}{2} \sqrt{x}\)[/tex] will also start from 0 and increase, as [tex]\(\sqrt{x}\)[/tex] increases.
3. Combining the insights:
- The smallest value of [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] is when [tex]\( x = 0 \)[/tex]. For [tex]\( x = 0 \)[/tex], [tex]\( f(0) = \frac{1}{2} \times \sqrt{0} = 0 \)[/tex].
- As [tex]\( x \)[/tex] increases beyond 0, [tex]\( \sqrt{x} \)[/tex] increases and therefore [tex]\(\frac{1}{2} \sqrt{x}\)[/tex] also increases, taking on all positive values.
Thus, the function [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] can take all non-negative real values.
Therefore, the range of the function [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] is all real numbers greater than or equal to 0.
Answer: The range is all real numbers greater than or equal to 0.
