Evaluate $f(-3), f(-2), f(-1)$ and $f(0)$

Expert intro: For a piecewise function, the first step is to check which rule applies to each input value. Boundary values like $x=-2$ need extra attention.

Detailed walkthrough

We use the definition

$$f(x)=\begin{cases} x+1 & \text{if } x<-2 \\ -2x-3 & \text{if } x\ge -2 \end{cases}$$

1) $f(-3)$

Since $-3<-2$, use $f(x)=x+1$:

$$f(-3)=-3+1=-2$$

2) $f(-2)$

Since $-2\ge -2$, use $f(x)=-2x-3$:

$$f(-2)=-2(-2)-3=4-3=1$$

3) $f(-1)$

Since $-1\ge -2$, use $f(x)=-2x-3$:

$$f(-1)=-2(-1)-3=2-3=-1$$

4) $f(0)$

Since $0\ge -2$, use $f(x)=-2x-3$:

$$f(0)=-2(0)-3=-3$$

Answers

$$f(-3)=-2,\quad f(-2)=1,\quad f(-1)=-1,\quad f(0)=-3$$

💡 Pitfall guide

The main mistake is using the wrong branch at the boundary. Because the second rule says $x\ge -2$, the value at $x=-2$ belongs to the second piece, not the first.

🔄 Real-world variant

If the breakpoint were written as $x>-2$ instead of $x\ge -2$, then $f(-2)$ would need a different rule or might be undefined at that point. Always check whether the boundary is included.

🔍 Related terms

piecewise function, substitution, boundary value