A piecewise function is a function defined by multiple subfunctions that
are each applied to separate intervals of the input.
LEARNING OBJECTIVES [ edit ]
Recognize gaps and jumps in piecewise functions
Solve for f(x), given x, in a piecewise function
KEY POINTS [ edit ]
Piecewise functions are defined using the common functional notation, where the body of the
function is an array of functions and associated subdomains.
The absolute value |x| is a very common piecewise function. For a real number, its value is ­x
when x < 0 and x when x ≥ 0 .
Piecewise functions may have horizontal or vertical gaps (or both) in their functions. A horizontal
gap means that the function is not defined for those inputs.
An open circle at the end of an interval in one of the subdomains means that the end point is not
included in the interval, i.e. strictly less than or strictly greater than. A closed circle means the end
point is included.
TERMS [ edit ]
piecewise function
A function that is defined by multiple subfunctions, with each subfunction applying to a certain
non­overlapping interval.
absolute value
For a real number, its numerical value without regard to itssign; formally, ­1 times the number if
the number is negative, and the number unmodified if it is zero or positive.
subdomain
A domain that is part of a larger domain.
Give us feedback on this content: FULL TEXT [ edit ]
In mathematics, a piecewise­defined
function, also called a piecewise function,
is a function which is defined by multiple
subfunctions, each subfunction applying
to a certain interval of the main function's
domain, a subdomain. Piecewise is
actually a way of expressing the function,
rather than a characteristic of the function
itself, but with additional qualification, it
can describe the nature of the function.
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Notation and Interpretation
Piecewise functions are defined using the common functional notation, where the body of the
function is an array of functions and associated subdomains. Crucially, in most settings,
there must only be a finite number of subdomains, each of which must be an interval, in
order for the overall function to be called "piecewise". For example, consider the piecewise
definition of the absolute value function:
|x| = {
−x,
x,
if x < 0
if x ≥ 0
For all values less than zero, the first function (−x) is used, which negates the sign of the
input value, making negative numbers positive. For all values of x greater than or equal to
zero, the second function (x) is used, which evaluates trivially to the input value itself .
Interactive Graph: Absolute Value
Graph of an absolute value. This is a piecewise function where $f(x)=x$ when $x\geq0$ and $f(x)=­x$
when $x\le0$.
Jumps
Piecewise functions can have jumps in either the input or theoutput.
If there are gaps in the input, then the function is not defined over those input values. For
example, the piecewise function: f(x) = {
−x,
x 2,
if x ≤ 0
if x ≥ 1
, x is not defined between 0 and
1, and there is a noticeable visible gap between the two parts of the function . Therefore the
domain of this function is x ≤ 0 and x ≥ 1 .
Interactive Graph: Gap in the Input
Graph of a piecewise function where there is a gap in the input. This piecewise function that defines
when , and when .
Piecewise functions can also have vertical gaps. An example of this is the piecewise function: f(x) = {
x 2,
2
if x < 1
−(x − 2) + 3,
2
−(1 − 2) + 3 = 2
if x ≥ 1
2
, has a vertical jump at x = 1. Since 1 = 1 and as seen by , not the open circle as on the first equation, x 2 . The open
circle means that it does not include that value, this is because the domain on the first
function is x < 1, strictly less than, therefore x does not equal 1. However, x can equal 0.9, or
0.99, or 0.9999, and so on, so there has to be some representation that it goes up to 1, but
never actually equals 1. A closed circle means that point is included, and the second part of
this equation does have a closed circle associated with it.
Interactive Graph: Vertical Jump of a Piecewise Function
Graph of a vertical jump in the piecewise function f(x)=\left\{\begin{matrix}x^{2}, & if\ x<1\\ ­(x­
2)^{2}+3, & if\ x\geq1\end{matrix}\right at .