To find the value of [tex]\( f(1) \)[/tex] for the given piecewise function, we need to carefully evaluate the function at [tex]\( x = 1 \)[/tex].
Given the piecewise function:
[tex]\[
f(x) = \begin{cases}
x^2 + 1, & -4 \leq x < 1 \\
-x^2, & 1 \leq x < 2 \\
3x, & x \geq 2
\end{cases}
\][/tex]
1. First, identify which interval [tex]\( x = 1 \)[/tex] falls into:
- The interval [tex]\([-4 \leq x < 1)\)[/tex] does not include [tex]\(1\)[/tex], since it is strictly less than 1.
- The interval [tex]\([1 \leq x < 2)\)[/tex] does include [tex]\(1\)[/tex], because [tex]\(1\)[/tex] lies within the range [tex]\(1 \leq x < 2\)[/tex].
- The interval [tex]\([x \geq 2)\)[/tex] does not include [tex]\(1\)[/tex], since [tex]\(1\)[/tex] is less than [tex]\(2\)[/tex].
2. [tex]\( x = 1 \)[/tex] falls into the interval [tex]\( [1 \leq x < 2) \)[/tex]. For this interval, the function is defined as:
[tex]\[
f(x) = -x^2
\][/tex]
3. Now substitute [tex]\( x = 1 \)[/tex] into this function definition:
[tex]\[
f(1) = -(1)^2 = -1
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{-1}\)[/tex].
