Piecewise-Defined Functions Many real-life applications cannot be modeled by one single function. Instead, their domain is broken into pieces; and a different function is used to model the application in each section of the domain. We call these âpiecewise-defined functions.â

A simple example involves income tax brackets. Letâs say that a state government imposes a 5% tax if your income is $5000 or less, and an 8% tax if your income is more than $5000. If your income is represented by x, we would define the tax function as

f(x) = 0.05x if x ⤠$5000

       0.08x if x > $5000

The graph of this function would appear as two discrete lines, with a disconnection at x = $5000.

 

Note that weâve used a closed circle at x = $5000 to denote that the lower line includes this value. Weâve also used an open circle at x = $5000 to denote that the upper line excludes this value.

A taxpayer making $4000 would lie on the lower segment, and pay $200; while a taxpayer making $8000 would lie on the upper segment, and pay $640.

In creating piecewise-defined functions, we are not restricted to linear segments. Piecewise-defined functions may consist of segments modeled by any type of function, including the quadratic, square root, or even reciprocal.

Example:

To discourage excessive driving, a proposed law imposes a tax on gasoline sales. If x represents the number of gallons purchased, the tax is defined as

f(x) = 0.10x for x ⤠20.0

       0.01x²Â for 20.0 < x ⤠40.0

       0.0006x³Â for x > 40.0

Graph this piecewise-defined function; and determine the amount of tax paid by motorists purchasing 15 gallons and 35 gallons of gas.

Solution:

A motorist purchasing 15 gallons would fall into the first segment; so the tax would be 0.1 · 15 = $1.50.

A motorist purchasing 35 gallons would fall into the second segment; so the tax would be 0.01 · (35)²Â = $12.25.

The graph is as follows:

 

 

 

13. Income tax in a particular country is calculated according to the following rate schedule: 15% on the first $19,000 of income 31% on the next $46,000 of income 49% on any further income

a. How much income tax should be paid by someone making $61,300?

b. How much income tax should be paid by someone making $101,900?

c. Construct a piecewise function that gives the tax for $x of income.

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