To determine the values of the piecewise function at specific points, let's analyze the given function step-by-step and evaluate it based on the given conditions.
The piecewise function is defined as follows:
[tex]\[
\begin{aligned}
+2x, & \quad \text{if } x \leq -1 \\
+\frac{3x}{2}, & \quad \text{if } -1 < x < 3 \\
\frac{1}{4}x, & \quad \text{if } x \geq 3
\end{aligned}
\][/tex]
Now, we need to evaluate this function at [tex]\(x = -3\)[/tex], [tex]\(x = -1\)[/tex], and [tex]\(x = 3\)[/tex].
### 1. Evaluating [tex]\(f(-3)\)[/tex]
Since [tex]\(-3 \leq -1\)[/tex], we use the first piece of the function:
[tex]\[
f(x) = 2x
\][/tex]
Plugging in [tex]\(x = -3\)[/tex]:
[tex]\[
f(-3) = 2(-3) = -6
\][/tex]
So, [tex]\(f(-3) = -6\)[/tex].
### 2. Evaluating [tex]\(f(-1)\)[/tex]
Since [tex]\(-1 \leq -1\)[/tex], we use the first piece of the function again:
[tex]\[
f(x) = 2x
\][/tex]
Plugging in [tex]\(x = -1\)[/tex]:
[tex]\[
f(-1) = 2(-1) = -2
\][/tex]
So, [tex]\(f(-1) = -2\)[/tex].
### 3. Evaluating [tex]\(f(3)\)[/tex]
Since [tex]\(3 \geq 3\)[/tex], we use the third piece of the function:
[tex]\[
f(x) = \frac{1}{4}x
\][/tex]
Plugging in [tex]\(x = 3\)[/tex]:
[tex]\[
f(3) = \frac{1}{4}(3) = 0.75
\][/tex]
So, [tex]\(f(3) = 0.75\)[/tex].
Therefore, the evaluated values of the function at the given points are:
[tex]\[
\begin{aligned}
f(-3) &= -6 \\
f(-1) &= -2 \\
f(3) &= 0.75
\end{aligned}
\][/tex]
