To determine the value of [tex]\( f(0) \)[/tex] for the piece-wise function, we need to evaluate the function definition at [tex]\( x = 0 \)[/tex].
The given piece-wise function is:
[tex]\[
f(x) = \begin{cases}
2x & \text{if } x < 1 \\
\frac{1}{2}x + \frac{5}{2} & \text{if } x \geq 1
\end{cases}
\][/tex]
Since [tex]\( 0 < 1 \)[/tex], we use the first piece of the function to find [tex]\( f(0) \)[/tex]. According to the first piece of the function:
[tex]\[
f(x) = 2x \quad \text{for} \quad x < 1
\][/tex]
Substitute [tex]\( x = 0 \)[/tex] into this piece:
[tex]\[
f(0) = 2 \cdot 0
\][/tex]
Doing the multiplication:
[tex]\[
f(0) = 0
\][/tex]
Therefore, the value of [tex]\( f(0) \)[/tex] is:
[tex]\[
f(0) = 0
\][/tex]
