Let's say that the first $15,000 you earn is taxed at a rateof 20%, Our income tax is based on a graduated tax calculation. Forexample, we can talk about "flat" income tax versusa "graduated" income tax.Ī flat income tax would tax people at the same rate regardlessof their income.įor 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. Why study piecewise functions? Well, there are some real-lifepractical examples for studying piecewise linear functions. Our range runsfrom because we have no negative outputsfor the function. 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. Graph of the absolute value function: y = |x| Now, let's graph this function using the points the chart aboveto plot our coordinates: Let's make a chart, substituting values in for x and solvingfor y = |x| as illustrated below. 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. So, whether x is positive, negative, or zero. The expression |x| is read "the absolute value of x." Graph of the Piecewise Function y = -x + 3 on theinterval Ī special example of a piecewise function is the absolute valuefunction that states: 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). For example, the graphof y = -x + 3 on the interval and the graph y = 3x + 1on the interval. Some piecewise functions are continuous like the one depictedabove, whereas some are not continuous. 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). In the first piece, the slopeis 2 or 2/1, while in the second piece, the slope is 0. Notice that the slope of the function isnot constant throughout the graph.
The graph depicted above is called piecewise because it consistsof two or more pieces. Graph of the piecewise function y = 2x + 3 on theinterval (-3, 1) Since the graphs do not includethe endpoints, the point where each graph starts and then stopsare open circles
Note that theyspan the interval from (-3, 5). 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.
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