Evaluate f(x) when x = -3, x = 2, and x = 4. Then graph f(x).

Evaluating a piecewise function adds an extra step to the whole proceedings. We have to decide which piece of the function to plug-and-chug into. Since -3 is less than 2, we use the first function to evaluate x = -3.

f(x) = x + 1

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

The number 2 is our boundary between life, death, and the two pieces of our function. Tie-breakers go to the second function, though.

f(x) = -2x + 7

f(2) = -2(2) + 7 = 3

The second function continues to be used, from 2 onward to infinity—and beyond, according to some space-faring toys.

f(x) = -2x + 7

f(4) = -2(4) + 7 = -1

Now, to graph the function.

To the left of x = 2, f(x) = x + 1. The graph will go right up to, but not touch, f(2) = 2 + 1 = 3. Then f(x) = -2x + 7 to the right of and including x = 2. We can also use the points we evaluated as guides.

Graph the function:

Careful, Shmooper. Every other time that we've done a piecewise function, the "if" conditions have started with the negative x-values and then become more positive as we go. That doesn't have to be the case, though. Always read the problem carefully, or you'll commit a terrible math blunder.

When x is less than -2, the function is a straight line at y = 2. At the end of the line, it's an inequality; the function doesn't equal 2 at x = -2. Instead, it equals:

f(-2) = 4(-2) – 4 = -12

The function equals 4x – 4 forever onward to the right after that.

Graph the piecewise function:

We'll just go one piece at a time, graphing each section in turn.

First, we have f(x) = x + 1, right up to, but not including, x = 3.

Then, at x = 3, the function just equals 3. Add a dot to the graph at (3, 3).

Finally, we add in f(x) = x + 2 for all points after x = 3, giving us the full graph.

This is still a function, because every value of x is still associated with one value of y. We just have to do some hunting at x = 3, seeing how it jumps around so much.