To determine the range of the function [tex]\( f(x) = \frac{H(x)}{2} \)[/tex] where [tex]\( H(x) \)[/tex] is the Heaviside function, let's start by understanding the Heaviside function itself.
The Heaviside function [tex]\( H(x) \)[/tex] is defined as follows:
[tex]\[
H(x) =
\begin{cases}
0 & \text{if } x < 0 \\
1 & \text{if } x \geq 0
\end{cases}
\][/tex]
Now, let's analyze the given function [tex]\( f(x) = \frac{H(x)}{2} \)[/tex]:
[tex]\[
f(x) = \frac{H(x)}{2} =
\begin{cases}
\frac{0}{2} = 0 & \text{if } x < 0 \\
\frac{1}{2} = \frac{1}{2} & \text{if } x \geq 0
\end{cases}
\][/tex]
From the above definition, we can see that:
- When [tex]\( x < 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex].
- When [tex]\( x \geq 0 \)[/tex], [tex]\( f(x) = \frac{1}{2} \)[/tex].
Thus, the function [tex]\( f(x) = \frac{H(x)}{2} \)[/tex] can only take on two different values depending on whether [tex]\( x \)[/tex] is negative or non-negative:
[tex]\[ f(x) \in \{0, \frac{1}{2}\} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\left\{0, \frac{1}{2}\right\}} \][/tex]
This corresponds to option (D). So, the range of the function [tex]\( f(x) = \frac{H(x)}{2} \)[/tex] is [tex]\(\left\{ 0, \frac{1}{2} \right\} \)[/tex].
