Okay, now when we are graphing piecewise functions we are really graphing several functions at once, except we are only going to graph them on very specific intervals. In this case we will be graphing the following two functions,

\[\begin{align*} - {x^2} + 4\hspace{0.25in} & {\mbox{on}}\hspace{0.25in}x < 1\\ 2x - 1\hspace{0.25in} & {\mbox{on}}\hspace{0.25in}x \ge 1\end{align*}\]

We’ll need to be a little careful with what is going on right at \(x = 1\) since technically that will only be valid for the bottom function. However, we’ll deal with that at the very end when we actually do the graph. For now, we will use \(x = 1\) in both functions.

The first thing to do here is to get a table of values for each function on the specified range and again we will use \(x = 1\) in both even though technically it only should be used with the bottom function.

\(x\) |
\( - {x^2} + 4\) |
\(\left( {x,y} \right)\) |

-2 |
0 |
\(\left( { - 2,0} \right)\) |

-1 |
3 |
\(\left( { - 1,3} \right)\) |

0 |
4 |
\(\left( {0,4} \right)\) |

1 |
3 |
\(\left( {1,3} \right)\) |

\(x\) |
\(2x - 1\) |
\(\left( {x,y} \right)\) |

1 |
1 |
\(\left( {1,1} \right)\) |

2 |
3 |
\(\left( {2,3} \right)\) |

3 |
5 |
\(\left( {3,5} \right)\) |

Here is a sketch of the graph and notice how we denoted the points at \(x = 1\). For the top function we used an open dot for the point at \(x = 1\) and for the bottom function we used a closed dot at \(x = 1\). In this way we make it clear on the graph that only the bottom function really has a point at \(x = 1\).

Notice that since the two graphs didn’t meet at \(x = 1\) we left a blank space in the graph. Do NOT connect these two points with a line. There really does need to be a break there to signify that the two portions do not meet at \(x = 1\).

Sometimes the two portions will meet at these points and at other times they won’t. We shouldn’t ever expect them to meet or not to meet until we’ve actually sketched the graph.