Explanation
A piecewise defined function is a function divided into branches by certain conditional rules. The conditions stand after the conditional "if". To evaluate a piecewise defined function for a particular value, we need to know first which condition it satisfies, and then, apply its corresponding rule.
Here, we have the function
[tex]f(x)=\begin{cases}x^2\text{ if }x<0 \\ x+2\text{ if }x\ge0\end{cases}\text{.}[/tex]
Let's begin the evaluations with -4. Note that it's less than 0, that is, it satisfies the first condition (x<0). By what I said before,
[tex]f(-4)=(-4)^2=-4\cdot-4=16.[/tex]
The next is f(-3). Again, -3<0. Thus
[tex]f(-3)=(-3)^2=-3\cdot-3=9.[/tex]
Now, 0 satisfies the second condition,
[tex]0\ge0.[/tex]
This means that we must apply the second rule (adding 2):
[tex]f(0)=0+2=2.[/tex]
The next two numbers 3 and 4 satisfy the second condition as well (for they are positive). Then,
[tex]f(3)=3+2=5,\text{ and }f(4)=4+2=6.[/tex]Answer[tex]\begin{gathered} f(-4)=16, \\ f(-3)=9, \\ f(0)=2, \\ f(3)=5, \\ f(4)=6. \end{gathered}[/tex]
