To find [tex]\( f(-3) \)[/tex] for the given piecewise function, let's follow the steps to evaluate the function at [tex]\( x = -3 \)[/tex].
The piecewise function is defined as:
[tex]\[
f(x) = \begin{cases}
2x + 2 & \text{if } x \leq 0 \\
- \frac{4}{3}x + 4 & \text{if } x > 0
\end{cases}
\][/tex]
Step 1: Determine which piece of the function to use based on the value of [tex]\( x \)[/tex].
We are given [tex]\( x = -3 \)[/tex].
Notice that [tex]\( -3 \leq 0 \)[/tex] is true. Therefore, we will use the first piece of the piecewise function:
[tex]\[
f(x) = 2x + 2
\][/tex]
Step 2: Substitute [tex]\( x = -3 \)[/tex] into the appropriate piece of the function.
[tex]\[
f(-3) = 2(-3) + 2
\][/tex]
Step 3: Perform the arithmetic operations.
First, multiply [tex]\( 2 \)[/tex] by [tex]\( -3 \)[/tex]:
[tex]\[
2 \cdot (-3) = -6
\][/tex]
Then, add [tex]\( 2 \)[/tex] to [tex]\( -6 \)[/tex]:
[tex]\[
-6 + 2 = -4
\][/tex]
Therefore, the value of [tex]\( f(-3) \)[/tex] is:
[tex]\[
f(-3) = -4
\][/tex]
So, [tex]\( f(-3) = -4 \)[/tex].
