To determine [tex]\( f(0) \)[/tex] for the given piecewise function, let's carefully follow each step to understand how to evaluate the function at [tex]\( x = 0 \)[/tex].
The given piecewise function is:
[tex]\[
f(x) = \begin{cases}
-x - 2 & \text{if } x < -2 \\
x + 4 & \text{if } x \geq -2
\end{cases}
\][/tex]
We need to evaluate this function at [tex]\( x = 0 \)[/tex].
Step-by-Step Solution:
1. Identify which part of the piecewise function to use by determining the value of [tex]\( x \)[/tex].
2. Since [tex]\( x = 0 \)[/tex], we need to check which condition [tex]\( x = 0 \)[/tex] satisfies:
- [tex]\( 0 < -2 \)[/tex]? No, this is false.
- [tex]\( 0 \geq -2 \)[/tex]? Yes, this is true.
3. The condition [tex]\( x \geq -2 \)[/tex] is satisfied, so we use the second part of the piecewise function:
[tex]\[
f(x) = x + 4
\][/tex]
4. Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
f(0) = 0 + 4
\][/tex]
5. Simplify the expression:
[tex]\[
f(0) = 4
\][/tex]
Thus, the value of [tex]\( f(0) \)[/tex] for the given piecewise function is [tex]\( 4 \)[/tex].
