To evaluate the piecewise function [tex]\( f(x) \)[/tex] at the given values of the independent variable, we need to follow the definition of the function for each specific range.
The piecewise function is defined as:
[tex]\[ f(x) = \begin{cases}
3x + 4 & \text{if } x < 0 \\
x + 7 & \text{if } x \geq 0
\end{cases} \][/tex]
Let's evaluate the function at each given value:
### (a) [tex]\( f(-2) \)[/tex]
For [tex]\( x = -2 \)[/tex]:
- Since [tex]\( -2 < 0 \)[/tex], we use the first piece of the function: [tex]\( f(x) = 3x + 4 \)[/tex].
- Substituting [tex]\( x = -2 \)[/tex] into this piece:
[tex]\[
f(-2) = 3(-2) + 4 = -6 + 4 = -2
\][/tex]
Thus, [tex]\( f(-2) = -2 \)[/tex].
### (b) [tex]\( f(0) \)[/tex]
For [tex]\( x = 0 \)[/tex]:
- Since [tex]\( 0 \geq 0 \)[/tex], we use the second piece of the function: [tex]\( f(x) = x + 7 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into this piece:
[tex]\[
f(0) = 0 + 7 = 7
\][/tex]
Thus, [tex]\( f(0) = 7 \)[/tex].
### (c) [tex]\( f(4) \)[/tex]
For [tex]\( x = 4 \)[/tex]:
- Since [tex]\( 4 \geq 0 \)[/tex], we use the second piece of the function: [tex]\( f(x) = x + 7 \)[/tex].
- Substituting [tex]\( x = 4 \)[/tex] into this piece:
[tex]\[
f(4) = 4 + 7 = 11
\][/tex]
Thus, [tex]\( f(4) = 11 \)[/tex].
To summarize:
- (a) [tex]\( f(-2) = -2 \)[/tex]
- (b) [tex]\( f(0) = 7 \)[/tex]
- (c) [tex]\( f(4) = 11 \)[/tex]
Therefore, the values of the function at the given points are:
- [tex]\( f(-2) = -2 \)[/tex]
- [tex]\( f(0) = 7 \)[/tex]
- [tex]\( f(4) = 11 \)[/tex]
