Everyday usage and use in mathematics
The word piecewise is not frequently used in everyday English. However, a common construction in the English language is to use the suffix "-wise" to indicate that something is being done in a specific manner. For example, if a piece of paper is folded lengthwise, this means that the paper is folded along a line parallel to the longer side of the paper. As another example, something moving clockwise is moving in the same direction as the hands of a clock.
In informal English, the suffix "-wise" can be attached to a noun or even a noun phrase. For example, a student who is doing well on the homework in a course but struggling with exams might say "Homeworkwise, this class is going well. Examwise, not so much."
In mathematics, a Piecewise function is a function that is defined differently at different parts of the domain. The domain is split up into pieces, and on each piece, the function has a different definition.
An example of a piecewise function is the function $f$ defined by \[ f(x) = \begin{cases} 5 & \text{if $x \leq 3$} \\ x + 4 & \text{if $x > 3$} \end{cases} \] Such a function is called piecewise because the definition of the function at a point $x$ in the domain is according to which piece of the domain $x$ belongs to. Imagine splitting the number line into two pieces: one piece where $x \leq 3$ and another piece where $x > 3$. The function has a different definition on each piece, so we call the function a piecewise function
Summary
The suffix "-wise" in English is used to used to indicate that something is done in a certain manner. In mathematics, a piecewise function is a function that is defined differently on different pieces of the domain.
Importance in mathematics
Piecewise functions are important in describing things that are subject to a sudden change. For example, imagine a function $g(t)$ that describes the height of a ball held from a window $19.6$ meters above the ground. Suppose that at time $t = 5$, the ball is released. Then the height of the ball might be given by the function \[ g(t) = \begin{cases} 19.6 & \text{if $t \leq 5$ } \\ 19.6 - 4.9 (t - 5)^2 & \text{if $5 < t < 7$ } \\ 0 & \text{if $t \geq 7$}. \end{cases} \] Before the time $t = 5$, the ball has not been released, so its height remains at $19.6$ meters. At the time $t = 5$, the ball is released, so it starts moving downward. At the time $t = 7$, the ball hits the ground, so it stops moving.
The absolute value function is the piecewise function defined by \[ |x| = \begin{cases} x & \text{if $x \geq 0$} \\ -x & \text{if $x < 0$}. \end{cases} \] The absolute value function is used throughout mathematics, particularly when describing distances.
Exercises
- Describe the use of the suffix "-wise" in English. Rephrase the following sentence without using "-wise": "Applewise, this was a good year for the orchard."
- A $60$ watt lightbulb is turned on at time $t = 2$ seconds and turned off at time $t = 8$ seconds. Give a piecewise function describing the brightness of the lightbulb at time $t$.
References
[1] Oxford English Dictionary