Piecewise Functions

Piecewise Function

1. Understand Piecewise-Defined Functions 

How is f(x) = 2|x| related to a linear function?  

Inspect the graph of the function f(x) = 2|x|. The graph has two pieces that meet at the vertex. Each piece is part of a line.  

Find the rule for each piece of the function.  

When x ≥0, the rule is f(x) = 2x.  

When x < 0, the rule is f(x) = -2x.   

You can write this function in terms of its pieces, each defined for a given domain.  

The function is a piecewise-defined function. A piecewise-defined function has different rules for different intervals of its domain.  

The function f(x) = 2|x| has two pieces over two intervals of the domain. For each interval, the piece is a linear function.  

2. Graph Piecewise-Defined Function 

1. What is the graph of: 

Graph each piece of the function for the given domain.  

2. Over what part of the domain is the function increasing? Decreasing? 

The function f increases when x ≤ 2 and decreasing when x > 2.  

3. Analyse the Graph of a Piecewise-Defined Function 

Cheyenne’s mother is reviewing the monthly water bills from the summer. Each monthly bill includes a graph like the one shown, which reflects the different rates charged for water based on usage.  

Several relatives visited Cheyenne’s family in July and their water bill is more than doubled. Assuming that the water consumption did not double that month, what is the possible explanation for the increase?  

The graph shows three tiers of pricing for water consumption: from 0 to 5000 gal, from 5001 to 10,000 gal, and more than 10,000 gal.  

The large increase in bill probably resulted from the usage increasing across one boundary of the domain, from Tier 1 to Tier 2, or Tier 2 to Tier 3.  

At the Tier 1- Tier 2 boundary 5,000 gal of usage results in a bill of $5 while using slightly more water results in a bill of at least $10.  

4. Apply a Piecewise-Defined Function  

A gym owner wants to purchase custom wristbands for marketing promotion. She thinks she will need about 75 bands. Her assistant insists that ordering over 100 wristbands will be less expensive than ordering 75. How can the assistant convince the gym owner?  

Formulate:  

Write a rule to represent each price point.  

Let x = number of wristbands 

Let f(x) = total cost  

Write a piecewise-defined function to represent the situation.  

Compute:  

Evaluate the function for f(75) and f(101).  

f(75) = 75 + 10 = 85  

f(101) = 0.5(101) = 50.5  

The cost for 75 wristbands is $85 and the cost for 101 wristbands is $50.5.  

Interpret:  

The gym owner will spend less if she orders more than 100 wristbands.  

Questions  

Question 1 

Express f(x) = -3|x| as a piecewise-defined function.   

Solution: 

Here is the graph of f(x) = -3|x|. 

If x ≥ 0, f(x) = -3x  

If x < 0, f(x) = 3x.   

Question 2 

Graph the following function:  

Solution:  

x = -2, f(x) = -4  

x = -1, f(x) = -3  

x = 0, f(x) = -2  

x = 1, f(x) = -1  

x = 2, f(x) = -1  

x = 3, f(x) = -3  

x = 4, f(x) = -5  

Question 3 

In the example explained in section 1.4, what is the difference in cost between one order of 200 wristbands, two orders of 100 wristbands each, and four orders of 50 wristbands each? 

Solution:  

x = 200  

f(x) = 0.5x = 0.5 × 200 = 100  

1 order of 200 wristbands costs $100.  

x = 100  

f(x) = x + 10 = 100 + 10 = $110  

2 orders of 100 wristbands cost $220.  

x = 50 

f(x) = 2x + 20 = 2 × 50 + 20 = $120  

4 orders of 50 wristbands cost $480.  

Exercise

Express the following functions as piecewise-defined functions:

1. f(x) = 0.6|x|
2. f(x) = (1/7) |x|
3. f(x) = Ix – 100I
4. f(x) = 3|x|
5. f(x) = 2 |x-1000|
6. f(x) = (1/3) |x-100|
7. f(x) = 4.5|x|
8. f(x) = 0.5|x+10|
9. f(x) = 10|x|
10. f(x) = 9|x+100|