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A piecewise function is useful when one situation cannot be described well by one formula. The domain is split into intervals, and each interval uses its own rule.
For example, electricity cost can be cheaper for the first usage tier and more expensive after a certain boundary. A boundary like this is a sign that the model should be written in pieces.
A piecewise function can be written in the form:
where are intervals that form a partition of the function's domain.
When reading or building a piecewise function, pay attention to three things:
A linear piecewise function uses a linear function on each interval. This form often appears when the rate of change is constant for a while, then changes after a certain point.
The function above can be written as:
The three rows are not three separate functions. They work together as one function, and the chosen formula depends on the value of .
Piecewise functions can also use quadratic pieces. This is helpful when one part of the model curves instead of following a straight line.
A piecewise function is continuous if the graph does not jump at a connection point. The value from the left, the value from the right, and the function value at that point must match.
Condition for continuity at point :
Example of a continuous piecewise function:
For continuity at :
Discontinuous piecewise functions have "jumps" or "holes" at certain points.
Many real-world situations can be modeled with piecewise functions because their rules change after certain boundaries. Progressive taxes, tiered parking fees, and tiered electricity rates are common examples.
Example: Electricity Rates
An electricity company applies tiered rates:
The mathematical model:
The number in the second row comes from the first . The number in the third row comes from the cost up to the first . This keeps the cost from restarting at when the usage enters a new interval.
Electricity cost table:
| Usage (kWh) | |||||
|---|---|---|---|---|---|
| Cost (Rp) |
Example: Multi-Modal Journey
Someone takes a journey with:
Distance function with respect to time:
The constants and keep the total distance moving forward from the last position, instead of returning to when the transportation mode changes.
To determine piecewise function equations from graphs or situations:
Example:
From a graph showing:
The solution can be summarized like this:
| Interval | Graph information | Equation |
|---|---|---|
| Passes through with slope | ||
| Horizontal line | ||
| Passes through with slope |
Piecewise function:
Determine the values of , , and for the function:
An online taxi company applies the following rates:
The base fare is for the first . Km costs . Above , the additional rate becomes .
Determine whether the following function is continuous at :
Sketch the graph of the function:
A worker is paid with the following system:
The first are paid at . Overtime from the th hour onward is paid at .
If the maximum work is , create a daily wage function!
Calculating function values:
For : since , use
For : since , use
For : since , use
Taxi fare model:
Let be the distance in km, then:
Checking continuity:
At :
Sketch of graph :
Daily wage function:
Let be the working hours, then:
Create a piecewise function model for the total cost!
Or simplified:
Since , the function is continuous at .
Or simplified:
