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Math 103: Precalculus12 chapters | 92 lessons | 10 flashcard sets

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

Piecewise functions are specific functions that have more than one piece. There is a special trick to graphing these type of functions, which you will learn in this lesson.

Mathematically, a **function** is a set of outputs related to specific inputs. Practically speaking, a function is like a machine. When you put something in, you get a specific something out. Let's say you have a candy machine - whenever you put a certain ingredient in, it makes a specific candy. If you add chocolate, you get fudge. If you add peanut butter, the machine makes cookies, and if you add fruit, out comes a pie. The machine works this way every time, no exceptions. You always get a specific outcome with each different input.

Mathematical functions work the same way. A **function** is an equation where each input gives a specific outcome. For example, *y = x + 2* is a function. For each input, *(x)*, you put in the function, you get a specific outcome, *(y).* An input of 2 gives an outcome of 4, always. You will never put a 2 into this particular function and get a different outcome.

A **piecewise function** is a function that has different parts, or pieces. The machine or the function works differently for each of the different pieces.

For example:

*f(x) = x - 2, x < 3*

*f(x) = (x - 1)^2, x is greater than or equal to 3*

This function behaves differently if the input is less than 3 or greater than or equal to 3. The most common piecewise function is the absolute value function. It works differently if the input is less than 0 than it does if the input is greater than 0.

The process of graphing a piecewise function is a bit different from graphing a regular function. The best way to graph a piecewise function is to think of the coordinate plane as a neighborhood and the functions as neighbors. First, we need to determine where the fence between the neighbors should be. In the case of our previous example, the fence goes at *x = 3*. So, the first step to graph this particular function is to draw a dotted vertical line at *x = 3*.

Next, you need to determine which neighbor owns the fence. This will always be the function with the 'equal to' part, in this case, *f(x) = (x - 1)^2*. This means that the graph of this line gets to sit on the fence, so it will be a closed dot on the line at *x = 3*. Finally, we can graph both equations separately on the correct side of the fence, remembering that the one with the equal sign gets to sit on the fence.

Let's try another example:

*f(x) = 4 - x, x is less than or equal to -2*

*f(x) = (x +1)/2, x > -2*

The fence line for this problem is at *x = -2*, and the equation *f(x) = 4 - x* is the equation that 'owns the fence.' The next step is to draw each line on the side of the fence where it belongs.

Piecewise functions do not occur only in pairs; there can be three or more equations in each function. Not every neighborhood has only two residents. Take a look at this example:

*f(x) = 1, x > 2*

*f(x) = 2 * x, x = 2*

*f(x) = -x^2, x < 2*

To graph this function, you use the same process as before, but this time, the fence is at *x = 2*, and the function that owns the fence is *f(x) = 2 * x*.

Start by graphing the top equation in its proper neighborhood. Then, place a solid dot at the point (2, 4), but no line coming from it since the function only occurs in that one spot.

Lastly, you can graph the last function, remembering that there is an open dot at *x = 2*.

**Piecewise functions** are functions that have more than one part, the most common of which is the absolute value function. Each piece of the function has a well-defined domain, or x-value. To graph a piecewise function, you need only to be sure that you have properly defined the area for each graph and then only place the graph in that area.

After this lesson, you'll be able to:

- Define functions and piecewise functions
- Recall the most common type of piecwise function
- Graph piecewise functions

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Math 103: Precalculus12 chapters | 92 lessons | 10 flashcard sets

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