Evaluate [tex]$f(3)$[/tex] for the piecewise function:

[tex]
f(x) = \left\{
\begin{array}{ll}
\frac{3x}{2} + 8, & x \ \textless \ -6 \\
-3x - 2, & -4 \leq x \leq 3 \\
4x + 4, & x \ \textgreater \ 3
\end{array}
\right.
[/tex]



Answer :

To evaluate \( f(3) \) for the given piecewise function, we need to determine which part of the piecewise function applies when \( x = 3 \). The piecewise function is defined as follows:

[tex]\[ f(x)=\left\{\begin{array}{ll} \frac{3 x}{2}+8, & x<-6 \\ -3 x-2, & -4 \leq x \leq 3 \\ 4 x+4, & x>3 \end{array}\right. \][/tex]

We need to find the part of the function that is valid for \( x = 3 \):

1. The first piece, \( \frac{3 x}{2} + 8 \), is valid when \( x < -6 \). Since \( 3 \) is not less than \( -6 \), this part does not apply.

2. The second piece, \( -3 x - 2 \), is valid when \( -4 \leq x \leq 3 \). Since \( 3 \) is within this interval (\(-4 \leq 3 \leq 3\)), this part of the function applies.

3. The third piece, \( 4 x + 4 \), is valid when \( x > 3 \). Since \( 3 \) is not greater than \( 3 \), this part does not apply.

Thus, we use the second piece of the function to evaluate \( f(3) \):

[tex]\[ f(x) = -3x - 2 \quad \text{for} \quad -4 \leq x \leq 3 \][/tex]

Substitute \( x = 3 \):

[tex]\[ f(3) = -3(3) - 2 \][/tex]

Now, perform the calculations:

[tex]\[ f(3) = -9 - 2 \][/tex]
[tex]\[ f(3) = -11 \][/tex]

Therefore, the value of the function \( f \) at \( x = 3 \) is:

[tex]\[ f(3) = -11 \][/tex]