Piecewise+Functions+-+B3

**What is a piecewise function?**
A piecewise function is a function which is defined by multiple subfunctions, each subfunction applying to a certain interval of the main function's domain. In other words, a piecewise function is a function made up of multiple subfunctions who have certain restrictions (x>0) that are put together to create the main function f(x). Here is an example:

How is a piecewise function like the functions previously studied?
A piecewise function is very similar to the other functions in that those previously studied functions are the subfunctions that make up a piecewise function. A piecewise function can be any combination of the other functions. However, a piecewise function is the only function where x cannot be defined from negative infinity to positive infinity on a single interval.

Shown above is an example of a piecewise function. You can see that it is made up of two different types of functions: y=-x/2 (polynomial function) and y=2sin(x) (trigonometric function). Even though the piecewise function is continuous, the domain needs to be split into two intervals (x≤0 and 0<x≤4) to describe the behavior of each section of the function.

As seen in the graph to the left, there are two functions graphed in the same window. This is the situation that makes up piecewise functions.

In order to write the piecewise function for this graph, you must first analyze the graph and look for recognizable functions. Then follow these steps... 1. After recognizing that there is a line and a parabola on the graph, you must determine if there are any dilations, or vertical/horizontal displacements. 2. As you can see, the parabola, f(x) = x2, has no displacement n'or dilation, but the line, f(x) = x + 7, has a vertical displacement of seven. 3. Next, look for where each function's restrictions are. In the graph above, the line, f(x) = x + 7, has no definite ending to the left, but stops when x = -2 on the right and the point is filled in. As you know from previous courses, a filled in point makes the inequality "less than or equal to" rather than just "less than." 4. By determining that f(x) = x + 7, when x ≤ -2, half of your piecewise function is complete! 5. Now look at the restrictions of the parabola. When x = -2, there is a point that is not filled in, making it simply greater than. But when continuing to the right, the parabola doesn't stop. 6. By determining that f(x) = x2, when x > -2, you have all the equations necessary to put together your piecewise function. 7. The final step is to make sure you format the piecewise function correctly, the correct way is shown on the very first picture...

How are the domain and range of a piecewise function determined?
Domain: All of the x-values for which the function is true. Range: All of the y-values for which the function is true.

1.) Use the inequality listed in the subfunction to find the domain. -use a bracket for greater/less than or equal to -use a parenthesis for greater than or less than 2.) combine the domains of the subfunctions into one domain. 3.) Find the range of each subfunction. 4.) Combine the ranges of the subfunctions into one range

Here is a video that will clarify finding the domain and range: media type="youtube" key="BxaYyS6lsQ4" width="425" height="350"

Why is the absolute value graph really a piecewise function?
Taking the absolute value of a negative number generates a positive answer. For this reason, the y-values of an absolute value graph are always positive. The parent function of all absolute value graphs is y=|x|. If you were to plug in 6 for x, the y-value would be 6. However, if you were to plug in -6 for x, the y-value would still be -6. For this reason, x-values on an absolute value graph are reflected across the y-axis, or over its vertex. The two sides of the graph have the same slope, however, on one side it is negative, and on the other it is positive. The different slopes make it a piecewise function.



Real-life applications of piecewise functions:
Because piecewise functions are applicable to all types of functions, such as linear, quadratic, absolute value, and so on, they appear in real-life mathematics often. One common real-life application of a piecewise function is in taxes. One often hears of a "flat income tax" compared to a "graduated income tax." A flat tax means that all incomes are taxed at the same rate. On a graph, where salary is the independent variable and taxes paid is the dependent variable, the slope remains constant as the income increases. With a graduated tax, different incomes have a different rate of taxation. This rate increases as income increases. This graph compares a flat tax, in blue, and a graduated tax, in pink:

As you can see, the slope of the blue line (flat tax) is constant. The slope of the pink line (graduated tax) changes. This makes the graduated tax an example of a piecewise function.

media type="youtube" key="fr-8tjLoeDw" width="425" height="350"If you scroll over the video at the end you can choose another video and watch him answer parts b and c!




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