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Piecewise defined functions, specifically the Absolute Value Function and a function representing the distance covered in an Ironman Triathlon as a function of elapsed time. It includes formulas, graphs, and examples.
Typology: Schemes and Mind Maps
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A Piecewise Defined Function Example A function may employ different formulas on different parts of itsdomain. Such a function is said to be piecewise defined. For example, the function graphed has the following formulas:
Functions Modeling Change: A Preparation for Calculus,4th Edition, 2011, Connally
for 2
for 2 6
2
(^) x x x y x
-2 0 2 4 6
y = x^2 y = 6 - x
x
y
Label (major) Endpoints!!!
The Absolute Value Function
Functions Modeling Change: A Preparation for Calculus,4th Edition, 2011, Connally
The Absolute Value Function is defined by
for 0 ( ) for^0 x x f x x x x
The Ironman Triathlon is a race that consists Text Example 3 of three parts: a 2.4 mile swim followed by a 112 mile bike race and then a 26.2 mile marathon. A participant swims steadily at 2 mph, cycles
steadily at 20 mph, and then runs steadily at 9 mph. Assuming that no time is lost during the transition from one stage to the next, find a formula for the distance d, covered in miles, as a function of the elapsed time t in hours, from the beginning of the race. Then Graph the function.
Text Example 3
t for x
t for x
t for t d SWIM
BIKE
RUN
Show how the equations and the end points generated?
t for x
t for x
t for t d
(d) Graph the cost function in part (a).
Graph the piecewise function
7 100
3 5
2 4 9 3 ( ) 2
1
2
x for x
x for x
x for x f x
K 0 A B C D E F G H I J TIME
RATE
Write appropriate units for each of the variables.
For each of the following statements give the largest
interval on which:
(a)
(c)
The number of people in the building is increasing fastest.
(d)
The number of
people in the building is decreasing.
