How to Graph Piecewise Functions: A Step-by-Step Guide

How to Graph Piecewise Functions: A Step-by-Step Guide

If you’re studying calculus or other advanced math topics, you may come across piecewise functions. These functions are essential for modeling real-world phenomena with multiple outcomes, and their graphs can help you understand the behavior of complex systems. In this article, we’ll explain what piecewise functions are, how to graph them, and provide examples that will help you master this concept.

Basics of Piecewise Functions

A piecewise function is a function that is defined by two or more sub-functions, with each piece only defined over a certain interval. These sub-functions are “pieced” together to form a single function that models the behavior of the system. Piecewise functions can be used to model any system with varying behavior, such as economic systems or physical systems.

To identify the domain and range of a piecewise function, you need to look at the intervals where each sub-function is defined. The domain of the function is the union of these intervals, and the range is the set of all possible y-values of the function.

Let’s take a look at some examples of piecewise functions:

Example 1:

$$
f(x)=\begin{cases}
x+2 &\text{for }x<0, \\ x^2 &\text{for }0\leq x\leq 2,\\ -x+4 &\text{for }x>2.
\end{cases}
$$

In this example, the function is defined by three sub-functions, each with their own interval of definition.

Example 2:

$$
g(x)=\begin{cases}
-x+5 &\text{for }x<3, \\ 2x-4 &\text{for }x\geq 3. \end{cases} $$

In this example, the function is only defined by two sub-functions, but they still have different intervals of definition.

Step-by-Step Tutorial on Graphing Piecewise Functions

Now that we know what a piecewise function is, let’s move on to graphing them.

The first step is to identify the intervals where each sub-function is defined. You should plot each sub-function separately, and then connect them to form the entire graph of the function.

Let’s take a look at how to graph Example 1 from above, $f(x)$:

1. Graph the function $x+2$ for $x<0$.
2. Graph the function $x^2$ for $0\leq x\leq 2$.
3. Graph the function $-x+4$ for $x>2$.
4. Connect the three subgraphs to form the complete graph of the piecewise function.

Here is what the complete graph should look like:


You can see that each sub-function is graphed separately and then connected to form the complete function.

Let’s try another example:

$$
h(x)=\begin{cases}
3x &\text{for }x\leq -2, \\
2 &\text{for }x>-2.
\end{cases}
$$

1. Graph the function $3x$ for $x\leq -2$.
2. Graph the function $2$ for $x>-2$. Since this is a constant function, the graph will be a horizontal line.
3. Connect the two subgraphs to form the complete graph of the piecewise function.

Here is what the complete graph should look like:


Again, notice how each sub-function is graphed separately before the two graphs are connected.

Try practicing on your own with some piecewise functions to become more comfortable with the process.

Visual Approach to Graphing Piecewise Functions

If you have trouble visualizing how to graph a piecewise function simply by looking at the equations, there are other methods you can use to help you. Diagrams and graphs can help you see the function more clearly and make it easier to graph.

One diagram that can be particularly helpful is the number line diagram. A number line diagram is a visualization of the intervals of the sub-functions used in the piecewise function. The diagram usually consists of a number line with the different intervals shaded different colors.

Let’s take a look at a number line diagram:


In this diagram, each interval is color-coded to correspond with the sub-function that is defined on that interval. This makes it easier to see which sub-function is defined where, and to graph each sub-function before connecting them.

You can also use graphs to help you visualize the function. For example, if you are graphing a piecewise function that involves absolute values, you can use the graphs of the absolute value function to help you.

Common Pitfalls in Graphing Piecewise Functions

Although graphing piecewise functions is a straightforward process, there are some common pitfalls you should avoid.

One common mistake is graphing each sub-function separately, but forgetting to connect them. Make sure that you connect the subgraphs to form the complete graph of the piecewise function.

Another common mistake is not identifying the intervals of definition correctly. Make sure to read the problem carefully and identify the intervals of definition for each sub-function.

Finally, be careful when graphing piecewise functions that involve absolute values. Sometimes these functions can be tricky because the absolute value function is not continuous, so you need to graph them piece-by-piece separately.

Applications of Piecewise Functions

Piecewise functions have applications in many fields, including physics and economics. The ability to model different systems and outcomes is essential for understanding how these fields work.

For example, in physics, piecewise functions can be used to model the motion of an object with changing speeds over time. In economics, piecewise functions might be used to model how a company’s revenue changes over time as the price of a product varies.

Being able to graph piecewise functions can make it easier to understand these models and see how different outcomes might arise.

Common Misconceptions about Graphing Piecewise Functions

Graphing piecewise functions can seem daunting at first, especially if you’re not used to dealing with them. However, with practice, you’ll find that it’s not as difficult as it seems.

One common misconception is that all piecewise functions are difficult to graph. While some functions can be more complex than others, you can usually break them down into smaller pieces and graph them separately to make the process easier.

Another misconception is that you need to be an expert in calculus to graph piecewise functions. While calculus can be helpful in some cases, you don’t need to be an expert to graph these functions. All you need is basic knowledge of functions and some practice.

Conclusion

Graphing piecewise functions can be a useful tool for modeling real-world phenomena with multiple outcomes. By breaking down the function into its sub-functions and graphing them separately, you can get a clear picture of how the system behaves.

We hope that this step-by-step guide has been helpful and that you’re now more comfortable graphing piecewise functions. Remember to always identify the intervals of definition, graph each sub-function separately, and then connect them to form the complete graph of the piecewise function.