To evaluate the piecewise function [tex]\( f(x) \)[/tex] at the given values, we need to follow the definition of [tex]\( f(x) \)[/tex] and substitute each value into the appropriate expression. The piecewise function is defined as:
[tex]\[
f(x) = \left\{
\begin{array}{ll}
4x + 4 & \text{if } x < 0 \\
3x + 6 & \text{if } x \geq 0
\end{array}
\right.
\][/tex]
Let's evaluate the function at the specified values:
(a) [tex]\( f(-4) \)[/tex]
Since [tex]\(-4 < 0\)[/tex], we use the expression [tex]\( f(x) = 4x + 4 \)[/tex]:
[tex]\[
f(-4) = 4(-4) + 4 = -16 + 4 = -12
\][/tex]
Therefore, [tex]\( f(-4) = -12 \)[/tex].
(b) [tex]\( f(0) \)[/tex]
Since [tex]\(0 \geq 0\)[/tex], we use the expression [tex]\( f(x) = 3x + 6 \)[/tex]:
[tex]\[
f(0) = 3(0) + 6 = 0 + 6 = 6
\][/tex]
Therefore, [tex]\( f(0) = 6 \)[/tex].
(c) [tex]\( f(1) \)[/tex]
Since [tex]\(1 \geq 0\)[/tex], we use the expression [tex]\( f(x) = 3x + 6 \)[/tex]:
[tex]\[
f(1) = 3(1) + 6 = 3 + 6 = 9
\][/tex]
Therefore, [tex]\( f(1) = 9 \)[/tex].
So the values of the function [tex]\( f(x) \)[/tex] at the given points are:
(a) [tex]\( f(-4) = -12 \)[/tex]
(b) [tex]\( f(0) = 6 \)[/tex]
(c) [tex]\( f(1) = 9 \)[/tex]
