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Piecewise Functions, a la Shmoop. John Lemon and Yoko Froyo run an unusually

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successful and longstanding lemonade and yogurt stand in a highly populated metropolitan area.

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It’s the end of a long day, and time to chart the daily earnings.

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From 6 AM to noon, John runs the stand.

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In order to attract people to the stand, he plays acoustic guitar and sings soothing,

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classic rock songs at no extra charge.

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He makes about 8 dollars an hour. Not bad for a lemonade and yogurt stand.

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John’s earnings can be represented by the equation f of x equals 8 “x,”…

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…where 8 dollars an hour times “x,”… the number of hours worked since 6 AM… will

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give us the monetary total.

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At noon, John leaves to work on his music, and Yoko takes over the stand until 6 PM.

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To attract people to the stand, she does various performance art pieces at no extra charge.

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Unfortunately, she doesn’t have as many fans as John. She makes about 5 dollars an

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hour.

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Yoko’s earnings can be represented by the equation g of y equals 5 “y,” where y

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is the number of hours worked since noon.

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How do we combine these two equations into one earnings chart? The answer lies in a piecewise

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function. We can create a comprehensive piecewise function

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that takes into account both John and Yoko’s total earnings over the course of the day.

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Here’s what the function looks like…

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F of “x” equals 8 “x” when “x” is greater than or equal to zero, and less

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than or equal to 6 since John works from 6 AM till noon...

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…AND 8 times 6… that’s John’s 48 dollars from his shift… plus 5 times… x minus

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6…

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…when “x” is greater than 6 and less than or equal to 12, since the stand closes

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at 6 pm, 12 hours after 6 AM.

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“x” is equal to the number of hours worked since 6 AM, which is why the second half of

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the function…

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…which represents Yoko’s earnings…

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…contains “x” minus 6.

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Her shift doesn’t start UNTIL noon, so the six hours before the switch don’t apply.

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And of course, F of “x” is the total earnings of the day.

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And now everything can be shown on one chart. All we do is graph the function.

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Both pieces are line segments, so two points per piece are sufficient to graph the function:

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For the first line segment for when “x” is greater than or equal to 0 or less than

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or equal to 6…

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…we can plug in x as 0, to get (0,0) as one endpoint.

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To get the other endpoint of the first segment, we can plug in x as 6 to get (6, 48).

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For the second line segment for when “x” is greater than 6 and less than or equal to

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12…

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…we can plug in "x" as 6 to get (6,48) to see where the second piece picks up from the

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first.

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To get the end point of the last piece, we can plug in 12 into the 8 times 6 plus 5 times…

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x minus 6…

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…to get 48 plus 5 times 6… 5 times 6 is 30, so 48 plus 30 equals 78.

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So our endpoint for the second line segment is (12,78).

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Perfect, we can graph it now.

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Piecewise functions are just big functions made out of smaller pieces of other functions.

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John loves them so much, he even has a song about them.

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