Step by step
The question
Question: graph f(x): 2x + 3 if x < 0, x² − 1 if x ≥ 0.
This is a piecewise function.
It has 2 rules.
Which rule we use depends on x.
Each x uses only one rule.
Careful!
At an edge, look at the sign.
Less than or greater than: the edge is not in, an open dot.
With "or equal to": a filled dot.
1 Make a table
Each rule gets its own x's, in its own colour.
The edge 0 we work out with both rules: it shows where each piece ends.
2 Draw the grid
Draw the grid.
3 Draw the first piece
2x + 3 is for x less than 0.
Plot its points.
At the edge 0: 0 is not in, so an open dot at (0, 3).
Join them with a straight line, only for these x's.
It keeps going on the open side.
4 Draw the second piece
x² − 1 is for x greater than or equal to 0.
Plot its points.
At the edge 0: 0 is in, so a filled dot at (0, −1).
Join them with a curve, only for these x's.
It keeps going on the open side.
5 Check
Let's check the edge x = 0.
Every x has only one y, so only one dot there can be filled.
0 belongs to the second rule.
f(0) = −1: the filled dot (0, −1).
It works!
Let's look back.
First: make a table.
Then: draw the grid.
Then: draw the first piece.
Then: draw the second piece.
Last: check.