To determine over which part of the domain the function [tex]\( f(x) \)[/tex] is increasing, we need to analyze the behavior of the function's pieces and their respective intervals.
The function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[
f(x) = \left\{
\begin{array}{l}
x + 3 \quad \text{for} \; x < -1 \\
-4x - 2 \quad \text{for} \; x \geq -1
\end{array}
\right.
\][/tex]
### Step-by-Step Analysis:
1. For [tex]\( x < -1 \)[/tex]:
Here, the function is defined as [tex]\( f(x) = x + 3 \)[/tex].
- To determine if this part is increasing, we'll consider the first derivative of [tex]\( f(x) = x + 3 \)[/tex]:
[tex]\[
\frac{d}{dx}(x + 3) = 1
\][/tex]
- Since the derivative is positive (1 > 0) for all [tex]\( x < -1 \)[/tex], the function is indeed increasing in this interval.
2. For [tex]\( x \geq -1 \)[/tex]:
Here, the function is given by [tex]\( f(x) = -4x - 2 \)[/tex].
- To determine if this part is increasing, consider the first derivative of [tex]\( f(x) = -4x - 2 \)[/tex]:
[tex]\[
\frac{d}{dx}(-4x - 2) = -4
\][/tex]
- Since the derivative is negative (-4 < 0) for all [tex]\( x \geq -1 \)[/tex], the function is decreasing in this interval.
### Conclusion:
From the analysis, we observe that the function [tex]\( f(x) \)[/tex] is increasing in the interval where [tex]\( x < -1 \)[/tex]. Therefore, the correct part of the domain over which the function [tex]\( f \)[/tex] is increasing is:
Option A: [tex]\( x < -1 \)[/tex]
