**By T. Barron & S. Kastberg**

Lesson 4

Piecewise Linear Functions

Consider the function y = 2x + 3 on the interval (-3, 1) andthe function y = 5 (a horizontal line) on the interval (1, 5).Let's graph those two functions on the same graph. Note that theyspan the interval from (-3, 5). Since the graphs do not includethe endpoints, the point where each graph starts and then stopsare open circles

**Graph of the piecewise function y = 2x + 3 on theinterval (-3, 1)**

**and y = 5 on the interval (1, 5)**

The graph depicted above is called piecewise because it consistsof two or more pieces. Notice that the slope of the function isnot constant throughout the graph. In the first piece, the slopeis 2 or 2/1, while in the second piece, the slope is 0. However,at the point where they adjoin, when we substitute 1 in for x,we get y = 5 for both functions, so they share the point (1, 5).

Some piecewise functions are continuous like the one depictedabove, whereas some are not continuous. For example, the graphof y = -x + 3 on the interval [-3, 0] and the graph y = 3x + 1on the interval [0, 3]. These functions do not share the samepoint at x = 0, as the first contains that point (0, 3), whilethe second piece contains the point (0, 1).

**Graph of the Piecewise Function y = -x + 3 on theinterval [-3, 0]**

**and y = 3x + 1 on the interval [0, 3]**

A special example of a piecewise function is the absolute valuefunction that states:

The expression |x| is read "the absolute value of x."

So, whether x is positive, negative, or zero,.Why? Well, in essence, the absolute value is a distance-measuringdevice and distance is always positive; even if you are walkingbackwards you are still going somewhere! The second part of thefunction seems confusing, because it seems like the answer shouldbe negative, but if x is less than zero to begin with, as it'sstated in the second part, then the answer is the opposite ofx, which is negative to begin with, so the answer is positive.

Let's make a chart, substituting values in for x and solvingfor y = |x| as illustrated below.

| |

| |

| |

| |

| |

| |

| |

| |

Now, let's graph this function using the points the chart aboveto plot our coordinates:

**Graph of the absolute value function: y = |x|**

Note that this piecewise linear function is continuous andit is in fact a function because it passes the vertical line test.Notice, also that the domain is becausewe can substitute anything real number in for x. Our range runsfrom because we have no negative outputsfor the function.

**A Real-Life Application**

Why study piecewise functions? Well, there are some real-lifepractical examples for studying piecewise linear functions. Forexample, we can talk about "flat" income tax versusa "graduated" income tax.

A flat income tax would tax people at the same rate regardlessof their income.

For instance, let's say that the flat tax is 30% of your income.Some people think that flat tax is unfair for those in or nearthe poverty level because they are getting taxed at the same rateas those in a higher income bracket.

Our income tax is based on a graduated tax calculation.

Let's say that the first $15,000 you earn is taxed at a rateof 20%,

the next $45,000 you make is taxed at a rate of 25%,

and any more money that you make above $45,000 would be taxedat a rate of 35%.

This would be an example of a piecewise continuos linear function.Let's take a look at the two graphs and discuss them.

**Graph of flat versus graduated taxes**

Note that the flat tax rate has a constant slope of .30, andthe equation used to find the amount of income tax paid is

**y = .30x,**

where x is the amount of money made, the independent variable,and y is the amount of income tax paid, the dependent variable.

For the graduated income tax (shown with the pink line), ifyou made less than or equal to $15,000 you would pay a constanttax rate of 20%.

So, the equation for this tax bracket is **y = .20x on theinterval [0, 15,000]**

with the same independent and dependent variables as indicatedabove.

For the second interval, from $15,000 to $45,000 you wouldpay a 25% tax rate. So, you would pay 20% on the first $15000,which would equal $3000 plus 25% on any amount over $15,000. Ifwe thought about this, we could develop the equation for thistax bracket as

**y = (.20)($15,000) + (.25)(x - $15,000)**

y = 3000 + .25x - 3750 (using the distributive property)

**y = .25x - 750 on the interval** [15,000, 45,000]

Note: Either equation above will work, the second one is justa simplified version of the first.

So, if you make $32,500, you could figure your taxes by theequation:

y = (.20)(15,000) + (.25)(32,500 - 15,000)

y = 3000 + (.25)(17500)

y = 3000 + 4375 = $7375

Using the second equation, you would get: y = .25(32,500) -750 = $7375.

Therefore, your total tax payment would be $7375.00, whereasin the flat tax of 30% your total tax payment would be y = (.30)(32,500)= $9750.00.

Now, in the last piece of the graduated tax, the income from[45,000 to "infinity"] is calculated as follows: Youwould be taxed 20% on the first $15,000 and 25% on the next $30,000.These are both explained above. Now, anything over $45,000 wouldbe taxed at 35%. So for the first $15,000, we pay (.20)(15,000)= 3,000. For the next $30,000 (on the inverval from [15,000, 45,000]we pay 25%, so we pay (.25)(30,000) = $7500. Any remaining incomeover $45,000 would be taxed at 35%, so we would pay (.35)(x -45,000), where x is our total income. So, we have a 3-piece equationfor our graduated taxes as follows:

y = .20x for [0, 15,000]

y = (.20)(15,000) + (.25)(x - 15,000) for [15,000, 45,000]

y = (.20)(15,000) + (.25)(30,000) + (.35)(x - 45,000) for [45,000to "infinity'].

Calculate what you would pay in taxes, both flat and graduated,if your salary is $77,000. What about if your salary is $160,000?

From the graph above, we see that the flat tax is worse forpeople in the lower income, as the flat tax line is above thegraduated tax line. However, note the characteristic of the graphsas income increases.

For which incomes(s) would the flat tax and the graduated taxbe the same? The answer is $105,000. We will discuss how to arriveat the algebraically in the next section, "Linear Systems."So, knowing something about piecewise functions may help you decidewhether or not to vote for graduated or flat income tax basedon your income! Note that these percentages are fictitious, soif you are planning to make a real-life decision, make sure youknow the correct tax percentages!

Back