1.6 Piecewise Functions
Objective:
•
Graph piecewise functions
•
Apply piecewise functions in a real world situational context
Warmup
With a partner, and having no prior guidance from me, try to graph the function below.
⎧x +1
f ( x) = ⎨ 2
⎩x
x<0
y
x≥0
x
Summarize Below
What worked?
What did you do incorrectly that you want to avoid going forward?
Ex 1: Graph.
⎧x + 5
⎪
f ( x ) = ⎨ −2 x − 1
⎪2 x − 9
⎩
Key Points:
x ≤ −2
−2 < x ≤ 2
x>2
y
x
Ex 2: Graph.
⎧2 x − 3
⎪
f ( x ) = ⎨3
⎪5
⎩
x<0
x=0
x>0
y
x
Ex 3: Graph.
⎧− x 2
⎪⎪
f ( x) = ⎨ x
⎪x − 4
⎪⎩
Key Points:
x<0
0≤ x<4
x>4
y
x
Can you go backwards?
Write the equation for the piecewise defined functions shown below.
⎧ _____
⎪
f ( x ) = ⎨ _____
⎪ _____
⎩
_________
_________
_________
Closure #1
A function and graph are shown below. Can you identify ALL the errors?
Closure #2
Write the equation for the piecewise defined functions shown below.
⎧ _____
f ( x) = ⎨
⎩ _____
_________
_________
Real World Applications
Precalculus in the Post Office
Postal charges for mailing packages depend on both weight and destination and this leads to an application of
piecewise functions. For example, the rates for a certain destination are shown in the table below.
Weight in pounds (x)
Postage Cost (y)
Under 1
$0.80
1 or more, but under 2
$1.00
2 or more, but under 4
$1.25
4 or more
$1.50
1) Create the piecewise function using the table above.
2) Graph the function.
y
x
Precalculus in your Taxes
In Democratopia, the leaders love to tax the villagers. Emperor Corzinithian has proposed the following tax
schedule for the year. Any village earnings up to and including $18,000 will be taxed at a rate of 15%, the
next $28,500 in earnings will be taxed at a rate of 28%, and all earnings beyond this amount will be taxed at a
rate of 32%.
(A) Construct a function, T(x), where x represents the income in dollars, which models the tax proposed by
the emperor.
(B) Graph T(x).
(C) Find the amount of taxes paid for an annual village income of $15,000, $25,000 and $50,000.
(D) What is the approximate income of a villager that paid $6,000 in taxes to Democratopia?
y
x