Find [tex]\( f(-3) \)[/tex] for the piecewise function.

[tex]\[
f(x)=\left\{\begin{array}{ll}
2x + 2 & \text{if } x \leq 0 \\
-\frac{4}{3}x + 4 & \text{if } x \ \textgreater \ 0
\end{array}\right.
\][/tex]

[tex]\[ f(-3) = [?] \][/tex]



Answer :

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].