FunctionsPiecewise Functions

Draw a Function:

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Multisport races test athletes endurance. Swimrun is a rather new multi sport competition that started in 2002 in Sweden. The story goes that the owner of the Utö Värdshus hotel, his friend, and two hotel staff challenged each other to a two-versus-two race from the Utö Värdshus hotel, across three islands, to Sandhamn. The losing team would pay for everyone’s post-race meals. How long do you think the race lasted?

The race ended up taking over 24 hours! The friends did the same race the next year, and the idea for the ÖtillÖ (island to island) was born.

This is an example of a piecewise function where different rules apply to different sets of input values. We can see the first section of the graph has a different slope than the second section.

One of the most common ways to write piecewise functions is by using cases.

dt=120t,0≤t<10,16t−76,10≤t≤40:

Each line in this function is a case. It includes the function rule and the input values where the rule is used. We read this function as “The function d has a value of (1/20)t when t is at least 0 and up to 10. Function d is (⅙)t-(7/6) when t is at least 10 and no more than 40.”

Let’s continue to get our feet wet in the world of piecewise functions.

Recall that functions cannot have one input going to more than one only one output value. The vertical line test is a tool to test whether a relation is a function. Use the vertical line above to test this relation.

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We need to pay close attention to the endpoints of each section of the domain. The endpoints ensure each element of the domain is matched to only one element of the range. With this in mind, select the function that matches the graph above.

option 1={(1.3, 0<=d<=100),(1.2, 100<d<=200),(1.4, 200<d<=300),(1.5, 300<d<=400):}) option 2={(1.3, 0<=d<=100),(1.2, 100<=d<=200),(1.4, 200<=d<300),(1.5, 300<d<=400):})

Notice the graph shows “<” as an open circle - the same would be true for endpoint containing “>”. The closed circles inculcate “<=” and “>=”.

The function s(d) is a special kind of piecewise function called a step function. One major difference between s(d) and d(t) above is all the slopes in s(d) are .

The fastest leg of the relay is freestyle butterfly breaststroke backstroke with a speed of meters per second. The slowest leg was breaststroke freestyle butterfly backstroke completed in meters per second. Each leg of the race was meters long.

We’re ready to dive into graphing.

One of the most common multisport competitions is a triathlon where athletes swim, bike, and run. The function l(t), Lisa Laws’s race, is given below. Use the given line segments to draw l(t) on the coordinate plane.

lt=75x,0≤t≤20,50000,20<t≤21,−615.385t+63000,21<t≤86,10000,86<t≤87,−277.778t+34166.7,87<t≤123:

triathlon transition

piece 1 piece 2 piece 3 piece 4 piece 5 solution

Noticing that each slope is either constant or positive negative helps us determine the orientation of each piece of the graph. Constant slope is a horizontal vertical line. Positive slope moves up down as we read from left to right.

Slope can also help us determine the order of the pieces from left to right. For example, Law’s fastest leg of the race was cycling swimming running. The largest slope, meters per minute, is the third case in the function. It runs between and minutes. We now know where on the x-axis to place the steepest piece of the graph.

Recall that a function’s key features give us insights into what’s going on during the race. For example, the starting line is represented by the {.FIXME} (multiple select) y-intercept x-intercept maximum minimum. We can write this point in function notation as l()=. Place the remaining statements on the graph.

| Place contextual statement cards on graph | | Target key feature appears when card is placed | |Function notation appears when card is placed | | | | | | | | Law crosses the finish line. | | maximum | | l(123)=51500 | | Law is cycling toward the transition point. | | increasing | | 21 < t <= 86 | | Law is transitioning from swimming to cycling. | | constant | | 20 < t <= 21 | | Law is transitioning from cycling to running. | | constant | | 86 < t <= 87 |

Now you get to race Law. One of the exciting things about triathlons is that you don’t need to be the fastest at each of the three sports, you just need to cross the finish line first. Here you can adjust your graph, s(t), to see how the race changes. Let’s say your most challenging leg of this race is swimming. As you can see, this segment cannot be adjusted. Can you beat Law with a swim leg that’s minutes slower?

Try adjusting the graph so that your swim and run are slower than Law’s. You need to cycle at meters per minute in order to beat Law.

graph mock-up

As you can see, your choice will depend on a few different factors. Fill out the table below to have a clearer understanding of your options.

| Weight | | Cost | | Cost | | | | | | | | 50 | | {.fixme} also accept NA, N/A, na, n/a, none, no | | 0.10 | | 75 | | | | | | 125 | | | | | | 150 | | | | | | 175 | | | | | | 225 | | | | | | 275 | | | | | | 325 | | | | | | 335 | | | | |

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