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Math 103: Precalculus11 chapters | 81 lessons

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

A piecewise function is a certain type of function that is made up of different parts. This lesson will define and describe piecewise functions and teach you how to identify them and how to write them.

When we talk about a person's function, we are talking about their job or role, the thing that they do. In math, a **function** is defined as a set of outputs related to specific inputs. This really just means that a function is an equation with a specific job to do. When you input a certain number into the equation, you will get a specific and unique number from the equation. Just like, for example, the function of the letter carrier. They get the mail (input) and deliver it to the proper address (output).

There are many different types of functions, just as there are multiple different types of career options. One type of mathematical function is called a **piecewise function**. A piecewise function is a function that has different parts, or pieces. Each part of the piecewise function has its own specific job that it performs when the conditions are correct.

For example:

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

*f*(*x*) = (*x* - 1)2, *x* â‰¥ 3

This function behaves differently if the input is < 3 than it does if the input is â‰¥ 3. The most common piecewise function is the absolute value function.

Here are some more examples of piecewise functions. As you can see, they do not have to be limited to only two equations, they can have many parts.

For example:

*f*(*x*) = 4 - *x*, *x* â‰¤ -4

*f*(*x*) = (*x* + 2)/2, *x* > -4

Or how about:

*f*(*x*) = -3, *x* > 1

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

*f*(*x*) = *x*/2, *x* < 1

And how about this one:

*f*(*x*) = -*x*, *x* < 0

*f*(*x*) = x2, 0 â‰¤ *x* < 4

*f*(*x*) = *x* - 5, 4 â‰¤ *x* < 6

*f*(*x*) = *x*/2, *x* â‰¥ 6

At some point, you may need to **evaluate a piecewise function**, that is, determine the output value when given certain input values. Take this function for example:

*f*(*x*) = 4 - *x*, *x* â‰¤ -4

*f*(*x*) = (*x* + 2)/2, *x* > -4

If you were asked to evaluate this function when *x* = 2 and *x* = -5, where would you start?

This first thing to do is determine which function describes *x* at 2 and which one describes *x* at -5.

By looking at the equations, we see that when *x* > -4, the equation used is *f*(*x*) = (*x* + 2)/2. So, for our first number *x* = 2, this is the equation we will use since 2 > -4. Solve this equation by plugging 2 in for *x* to get *f*(2) = (2 + 2)/2 = 4/2 = 2. The first evaluation gives us the point (2, 2).

For the next number *x* = -5, we want to use the equation *f*(*x*) = 4 - *x* because -5 < -4. Solving this equation gives us *f*(-5) = 4 - (-5) = 4 + 5 = 9. This gives us the point (-5, 9).

Piecewise functions are not just mathematical exercises. They do have practical applications. Take this example.

Your dog groomer charges you based on the weight of your dog. If your dog is 40 pounds or less, she charges $30 for a wash, but if your dog is over 40 pounds, she charges $30 plus $2 for every pound over 40. Write a piecewise equation to describe her fees. What would she charge to wash a 65 pound Dalmatian?

The first step is to find the boundary lines. This particular scenario has one boundary line where the price rate changes at 40 pounds. Because there is only one boundary line, we know there will be only two equations. One equation describes the rate 40 pounds and below, and the other equation describes the rate above 40 pounds.

Let's start by writing the equation for below 40 pounds. Since it is a flat rate, we know that the line will be a straight horizontal line at 30. The boundary is from 0 (since a dog cannot weigh under 0 pounds) to 40, and the equation is:

*f*(*x*) = 30, 0 â‰¤ *x* â‰¤ 40

The other section describes the rate for dogs that weigh over 40 pounds. That equation looks like this:

*f*(*x*) = 2(*x* - 40) + 30, *x* > 40

So, the piecewise function for this scenario looks like this:

*f*(*x*) = 30, 0 < or = *x* â‰¤ 40

*f*(*x*) = 2(*x* - 40) + 30, *x* > 40

To find the cost for washing a 65 pound Dalmatian, all you need to do is determine which equation to use and solve it. Since 65 pounds is greater than 40, the equation you need to use is the second one. By plugging 65 in for *x*, you can determine the rate to wash the dog.

*f*(65) = 2(65 - 40) + 30 = 2(25) + 30 = 50 + 30 = 80

So, the cost to get my dog washed will be $60.

Piecewise functions can also be used to find rates for any number of goods and services, as well as other scientific and financial applications.

Piecewise functions are functions that have multiple parts. Each part is defined by a specific domain. These functions can be used to calculate payment rates and other scientific and financial problems.

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Math 103: Precalculus11 chapters | 81 lessons

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