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Study Guides > ALGEBRA / TRIG I

Writing Piecewise Functions

Learning Outcomes

  • Define piecewise function
  • Evaluate a piecewise function
  • Write a piecewise function given an application

A piecewise function is a function where more than one formula is used to define the output over different pieces of the domain.

We use piecewise functions to describe situations where a rule or relationship changes as the input value crosses certain "boundaries." Piecewise function are useful in many real-world situations.  For example, we often encounter situations in business where the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system where incomes up to $10,000 are taxed at 10%10\% and any additional income is taxed at 20%20\%. The tax on a total income, S, would be 0.10.1S if S≤S\le $10,000 and 1000 + 0.2 (S - $10,000) if S > $10,000.

Piecewise Function

A piecewise function is a function where more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all of these smaller domains. We notate this idea like this:

f(x)={formula 1 if x is in domain 1formula 2 if x is in domain 2formula 3 if x is in domain 3 f\left(x\right)=\begin{cases}\text{formula 1 if x is in domain 1}\\ \text{formula 2 if x is in domain 2}\\ \text{formula 3 if x is in domain 3}\end{cases} In piecewise notation, the absolute value function is
∣x∣={x if x≥0−x if x<0|x|=\begin{cases}x\text{ if }x\ge 0\\ -x\text{ if }x<0\end{cases}

Evaluate a Piecewise-Defined Function

In the first example, we will show how to evaluate a piecewise defined function. Note how it is important to pay attention to the domain to determine which expression to use to evaluate the input.

Example

Given the function f(x)={7x+3 if x<07x+6 if x≥0f(x)=\begin{cases}7x+3\text{ if }x<0\\7x+6\text{ if }x\ge{0}\end{cases}, evaluate:
  1. f(−1)f (-1)
  2. f(0)f (0)
  3. f(2)f (2)

Answer: In order to evaluate a function for a given xx value, we first have to determine which domain that xx values falls into.  In this example, if the xx value is less than zero then we will use the first formula.  If the given xx value is greater than or equal to zero, then we will use the second formula. 1. f(x)f(x) is defined as 7x+37x+3 for x=−1 because −1<0x=-1\text{ because }-1<0.   Evaluate: f(−1)=7(−1)+3=−7+3=−4f(-1)=7(-1)+3=-7+3=-4 2. f(x)f(x) is defined as 7x+67x+6 for x=0 because 0≥0x=0\text{ because }0\ge{0}. Evaluate: f(0)=7(0)+6=0+6=6f(0)=7(0)+6=0+6=6 3. f(x)f(x) is defined as 7x+67x+6 for x=2 because 2≥0x=2\text{ because }2\ge{0}. Evaluate: f(2)=7(2)+6=14+6=20f(2)=7(2)+6=14+6=20

In the following video, we show how to evaluate several values given a piecewise-defined function. https://youtu.be/E2F2-gP-2qU In the next example, we show how to evaluate a function that models the cost of data transfer for a phone company.

Example

A cell phone company uses the function below to determine the cost, CC, in dollars for gg gigabytes of data transfer.

C(g)={25 if 0<g<210g+5 if g≥2C\left(g\right)=\begin{cases}{25}\text{ if }{ 0 }<{ g }<{ 2 }\\ 10g+5\text{ if }{ g}\ge{ 2 }\end{cases}

Find the cost of using 1.51.5 gigabytes of data and the cost of using 44 gigabytes of data.

Answer:

To find the cost of using 1.51.5 gigabytes of data, C(1.5)(1.5), we first look to see which part of the domain our input falls. Because 1.51.5 is less than 22, we use the first formula.

C(1.5) = $25

To find the cost of using 44 gigabytes of data, C(4)(4), we see that our input of 44 is greater than 22, so we use the second formula.

C(4)=10(4)+5=$45

The function from the previous example is represented in the graph below. We can see where the function changes from a constant to a line with a positive slope at g=2g=2. We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain. Graph of C(g)

 Write a Piecewise-Defined Function

In the last example, we will show how to write a piecewise-defined function that models the price of a guided museum tour.

Example

A museum charges $5 per person for a guided tour for a group of 11 to 99 people or a fixed $50 fee for a group of 1010 or more people. Write a function relating the number of people, nn, to the cost, CC.

Answer: Two different formulas will be needed. For n-values under 1010, C=5nC=5n. For values of n that are 1010 or greater, C=50C=50. C(n)={5n if 0<n<1050 if n≥10C(n)=\begin{cases}{5n}\text{ if }{0}<{n}<{10}\\ 50\text{ if }{n}\ge 10\end{cases}

A graph of the function for the previous example is shown below. The graph is a diagonal line from n=0n=0 to n=10n=10 and a constant after that. In this example, the two formulas agree at the meeting point where n=10n=10, but not all piecewise functions have this property. Graph of C(n). In the following video, we show an example of how to write a piecewise-defined function given a scenario. https://youtu.be/58mEZ4mEnUI

How To: Given a piecewise function, write the formula and identify the domain for each interval

  1. Identify the intervals where different rules apply.
  2. Determine formulas for the rules that describe how to calculate an output from an input in each interval.
  3. Use a bracket and "if" statements to write the function.

Summary

  • A piecewise function is a function where more than one formula is used to define the output over different pieces of the domain.
  • Evaluating a piecewise function means you need to pay close attention to the correct expression used for the given input.

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