Piecewise Function is a function that behaves differently for different types of input. As we know a function is a mathematical object which associates each input with exactly one output.
For example: If a function takes on any input and gives the output as 3. It can be represented mathematically as f(x) = 3. But in the case of the Piecewise function, it is defined by individual expressions for each interval.
A piecewise function is a function that is defined differently over different intervals of its domain. Instead of using a single equation for all inputs, it assigns distinct expressions to specific intervals.
Piecewise FunctionThe general piecewise function can be written mathematically as:
\bold{f(x) =\begin{cases} f_1(x),& \text{if } x < a \\ f_2(x),& \text{if } a \leq x < b \\ f_3(x),& \text{if } b \leq x \end{cases}}
Where,
- f1(x), f2(x), and f3(x) are three different functions, and
- a, b, and c are some real numbers.
The above expression for piecewise function means that for x less than a, the function takes on the value of f1(x), for x between a and b, it takes on the value of f2(x), and for x greater than or equal to b, it takes on the value of f3(x).
Domain and Range of Piecewise Function
Domain and Range of a piecewise function can be calculated using the domain and range of the individual pieces and taking the union of that range and domains.
Example: Find the Range and Domain of function f(x) which is defined as follows:
\bold{f(x) = \begin{cases} x, & x<1 \\ 2, & 1 \leq x \leq 5\\ x^2, & x>5 \end{cases}}
Solution:
As the function is defined for all the real numbers, so its domain is R if is defined for some portion of R then that portion becomes its domain.
Now for Range of function, for x<2, f(x) = x, thus range for this part is x<2
For 2 ≤ x ≤ 5, f(x) = 2, thus its range is 2 as it is a constant function for this interval.
For x > 5, f(x) = x2, f(5) = 25 and x2 is continuous and increasing function for x > 0, thus range is x>25.
Now, the union of all the ranges is {x<2} U {2} U {x>25} = (-∞, 2] U (25, ∞)
Piecewise Function Graph
To graph the Piecewise Function, we just need to graph the function individually for all the different intervals it is defined.
Example: Plot the graph of the function defined as follows:
\bold{f(x) = \begin{cases} x, & x<-1 \\ 2, & -1 \leq x \leq 2\\ x^2, & x>2 \end{cases}}
Solution:
As the domain of the function is the complete set of real numbers, thus there is no such values in R for which the function is not defined.
Now, for the first piece of graph for x < -1, is given as f(x) = x, which can be easily plotted. So the graph of a function for x < -1 is a straight line with a slope of 1 that passes through the origin.
For the second piece of the graph for -1 ≤ x ≤ 2, the given function is a constant function as f(x) = 2. So the graph of a function for -1 ≤ x ≤ 2 is again a straight horizontal line which is at a 2 unit distance from the x-axis.
For the third piece of the graph for x > 2, the given function is a parabolic curve that opens upwards and x2 is a increasing and continuous function, so the graph starts at the point (2, 4) goes in the upward direction as parabolic curve.
Plot these three pieces of the graph to obtain the required graph of the function.
Piecewise Function Examples
There are many famous examples of piecewise functions, some of which are as follows:
Modulus Function
The modulus function (f(x) = ∣x∣), also known as the absolute value function, is defined as:
\bold{f(x)= \begin{cases}x,& \text{if } x\geq 0\\-x,& \text{if } x<0 \end{cases}}
Outputs the absolute value of x.
Examples:
- f(5) = ∣5∣ = 5
- f(−3) = ∣−3∣ = 3
- f(0) = ∣0∣ = 0
The graph of absolute value function is as follows:
Floor Function
The floor function also called the greatest integer function or integer value, gives the largest integer less than or equal to x. The domain for this function is all the real numbers R while the range of this function is all the integers Z.
Examples:
- ⌊4.7⌋ = 4
- ⌊−2.3⌋ = −3
- ⌊6⌋ = 6
- ⌊0.99⌋ = 0
The graph of the floor function appears as a step function, with flat segments at each integer and jumps at every whole number:
Question 1: What is the Floor of 1.43?
Solution:
Floor of a number is the greatest Integer lesser or equal to that number. Therefore, here the Floor of 1.43 is 1.
Question 2: What is the Floor of -5.66?
Solution:
On Negative axis, the greatest Integer lesser than -5.66 is -6.
Hence, -6 is the Floor of -5.66.
Ceiling Function
This function returns the smallest successive integer. The ceiling function of a real number x is the least integer that is greater than or equal to the given number x. The domain for this function is R and range Z.
Similar to the floor function, the domain of the ceiling function is R and the range is all the integers I.
Example 1: What is the Ceiling of 1.43?
Solution:
The ceiling of the 1.43 should be its smallest successive Integer, hence the ceiling of 1.43 is 2.
Example 2: What is the ceiling of -7.8?
Solution:
The smallest successive Integer of -7.8 is -7.
Hence, -7 is the ceiling of -7.8.
Unit Step Function
Unit Step Function is yet another type of function used a lot in Signals and systems studies. It is defined as,
\bold{f(x) = \begin{cases}0,&x<0\\ 1,&x\geq0\end{cases}}
This function has no value at x = 0. It is called a step function because, at t = 0, it takes a step from 0 to 1. The domain for this function is R - {0} and range {0,1}.
Signum Function
This function shows the polarity of the input number if the number is negative function spits out -1 as output and if the number is positive the signum function spits out +1 as output and for 0 which is neutral in nature signum function spits 0 as output. Mathematically signum function is defined as
\bold{sgn(x) = \begin{cases} 1, & x>0\\ 0, & x = 0\\ -1, & x<0 \end{cases}}
The graph of the signum function is as follows:
Evaluating Piecewise Functions
Let's consider the following examples of piecewise functions to evaluate their value at any given point.
Example: Find the value of the following function at x = -2 and x = 10.
\bold{f(x) =\left\{\begin{matrix}x^{2},& \text{if } x\geq 0\\ -x,& \text{otherwise}\end{matrix}\right.}
Solution:
f(x)= \begin{cases} x^{2},& \text{if } x\geq 0\\ -x,& \text{otherwise} \end{cases}
at x = -2, x < 0 so the f(-2) = -2.
at x = 10, x > 0, so f(10) = 102 = 100.
Example: An arcade game charges the following prices depending on the length of time:
- Up to 6 minutes costs Rs.10
- Over 6 and up to 15 minutes costs Rs.15
- Over 15 minutes costs Rs.15 plus Rs.1 per minute above 15 minutes
Represent this as a piecewise function and tell the price charged if Anil played the game for 13 minutes and Raju played for 20 minutes.
Solution:
These kind of prices charges can be represented as,
f(t)= \begin{cases} 10, & \text{if } t\leq 6\\ 15 & \text{if } t \leq 15 \text{ and } t \gt 6 \\ 15 + 1(t - 15), & \text{otherwise} \end{cases}
at x = 13, f(13) = Rs.15 and x = 20, f(20 ) = 15 + 1( 20 – 15) = 20.
Piecewise Continuous Function
A Piecewise Continuous Function is a function that is continuous across its entire domain i.e., each piece of the function is continuous itself and all the intersection points are the same for each piece so where each piece ends another piece of function starts from there in the graph.
An example of a Piecewise Continuous Function is given as follows:
\bold{f(x) = \begin{cases} x^2,& x < 0\\ 2x,& x \geq 0\end{cases}}
This function is continuous across the entire domain because the limit of f(x) as x approaches 0 from the left (for x < 0) is equal to the limit of f(x) as x approaches 0 from the right (For x ≥ 0), and both limits are equal to 0. Therefore, f(x) is continuous at x = 0.
Piecewise Continuous Function Graph
A Piecewise Continuous Function Graph is given below. The graph of a piecewise continuous function often resembles a series of connected segments, each representing the function's behavior within a specific interval. At the points where intervals meet, the function may experience jumps, breaks, or other types of discontinuities.
f(x) = \begin{cases} x+3, & x<-1\\ 2, & -1\leq x \leq 2 \\ x^2 - 2, & x>2 \end{cases}
Read more: Continuity of Functions
Let's consider another example continuous piecewise function as follows:
\bold{f(x) = \begin{cases} x^2,& x < -1\\ x + 1,& -1 \leq x < 0\\ x - 1,& 0 \leq x < 1 \\ -x,& x \geq 1 \end{cases}}
This function is continuous across the entire domain as \lim_{x\to-1} f(x), \lim_{x\to0} f(x), and \lim_{x\to1} f(x), all exists.
Articles related to Piecewise Function:
Piecewise Function Worksheet
Question 1: Given the piecewise function f(x) = \begin{cases} x^2, & \text{if } x \leq 2 \\ 3x - 1, & \text{if } x > 2 \end{cases}. Find f(0)
Question 2: Evaluate the piecewise function g(x) = \begin{cases} 2x + 1, & \text{if } x < 0 \\ x^2, & \text{if } x \geq 0 \end{cases} at x = 3.
Question 3: Determine the domain of the piecewise function h(x) = \begin{cases} \sqrt{x}, & \text{if } x \geq 0 \\ \frac{1}{x}, & \text{if } x < 0 \end{cases}
Question 4: Find the value of k(2) for the piecewise function k(x) = \begin{cases} x^3, & \text{if } x \leq 1 \\ x^2, & \text{if } x > 1 \end{cases}
Question 5: Determine the range of the piecewise function n(x) = \begin{cases} x^2, & \text{if } x \leq 2 \\ 3, & \text{if } x > 2 \end{cases}
Similar Reads
Chapter 1: Sets
Representation of a SetSets are defined as collections of well-defined data. In Math, a Set is a tool that helps to classify and collect data belonging to the same category. Even though the elements used in sets are all different from each other, they are all similar as they belong to one group. For instance, a set of dif
8 min read
Types Of SetsIn mathematics, a set is defined as a well-defined collection of distinct elements that share a common property. These elements— like numbers, letters, or even other sets are listed in curly brackets "{ }" and represented by capital letters. For example, a set can include days of the week. The diffe
13 min read
Universal SetsUniversal Set is a set that has all the elements associated with a given set, without any repetition. Suppose we have two sets P = {1, 3, 5} and Q = {2, 4, 6} then the universal set of P and Q is U = {1, 2, 3, 4, 5, 6}. We generally use U to denote universal sets. Universal Set is a type of set that
6 min read
Venn DiagramVenn diagrams are visual tools used to show relationships between different sets. They use overlapping circles to represent how sets intersect, share elements, or stay separate. These diagrams help categorize items, making it easier to understand similarities and differences. In mathematics, Venn di
14 min read
Operations on SetsSets are fundamental in mathematics and are collections of distinct objects, considered as a whole. In this article, we will explore the basic operations you can perform on sets, such as union, intersection, difference, and complement. These operations help us understand how sets interact with each
15+ min read
Union of SetsUnion of two sets means finding a set containing all the values in both sets. It is denoted using the symbol '∪' and is read as the union. Example 1:If A = {1, 3. 5. 7} and B = {1, 2, 3} then A∪B is read as A union B and its value is,A∪B = {1, 2, 3, 5, 7}Example 2:If A = {1, 3. 5.7} and B = {2, 4} t
12 min read
Chapter 2: Relations & Functions
Chapter 3: Trigonometric Functions
Chapter 4: Principle of Mathematical Induction
Chapter 5: Complex Numbers and Quadratic Equations
Complex NumbersComplex numbers are an essential concept in mathematics, extending the idea of numbers to include solutions for equations that don't have real solutions. Complex numbers have applications in many scientific research areas, signal processing, electromagnetism, fluid dynamics, quantum mechanics, and v
12 min read
Algebraic Operations on Complex NumbersA complex number is a number that includes both a real and an imaginary part. It is written in the form:z = a + biWhere:a is the real part,b is the imaginary part,i is the imaginary unit, satisfying i2 = −1.Algebraic operations on complex numbers follow specific rules based on their real and imagina
7 min read
Absolute Value of a Complex NumberThe absolute value (also called the modulus) of a complex number z = a + bi is its distance from the origin in the complex plane. The absolute value tells you how far a number is from zero, regardless of its direction (positive or negative).It is denoted as ∣z∣ and is given by the formula:|z| = \sqr
7 min read
Conjugate of Complex NumbersIn the world of mathematics, complex numbers are one of the most important discoveries by mathematicians as they help us solve many real-life problems in various fields such as the study of electromagnetic waves, engineering, and physics.The Conjugate of a Complex Number is also a complex number obt
6 min read
Polar Representation of Complex NumbersComplex numbers, which take the form z = x + yi, can also be represented in a way that highlights their geometric properties. This alternative representation is known as the polar form. The polar representation of a complex number expresses it in terms of its magnitude (modulus) and direction (argum
9 min read
Imaginary NumbersImaginary numbers are numbers as the name suggests are the number that is not real numbers. All the numbers real and imaginary come under the categories of complex numbers. Imaginary numbers are very useful in solving quadratic equations and other equations whose solutions can not easily be found us
9 min read
Chapter 6: Linear Inequalities
Compound InequalitiesCompound Inequalities are the combination of two or more inequalities. These inequalities are combined using two conditions that are AND, and OR. These conditions have specific meanings and they are solved differently. The inequities in compound inequalities are individually solved using normal rule
10 min read
Algebraic Solutions of Linear Inequalities in One VariableA linear inequality is a mathematical expression involving an inequality symbol (<, >, ≤, or ≥) and a linear expression. Unlike linear equations, which give a specific solution, linear inequalities define a range of possible solutions.Example: 2x+3>5 In this case, the inequality indicates t
8 min read
Graphical Solution of Linear Inequalities in Two VariablesWe know how to formulate equations of different degree, and it is used a lot in real life, but the question arises, is it always possible to convert a situation into an equation? Sometimes we get statements like, the number of Covid cases per day in Delhi has reached more than 10,000. This phrase “L
8 min read
Solving Linear Inequalities Word ProblemsWe are well versed with equations in multiple variables. Linear Equations represent a point in a single dimension, a line in a two-dimensional, and a plane in a three-dimensional world. Solutions to linear inequalities represent a region of the Cartesian plane. It becomes essential for us to know ho
10 min read
Chapter 7: Permutations and Combinations
Chapter 8: Binomial Theorem
Chapter 9: Sequences and Series
Sequences and SeriesA sequence is an ordered list of numbers following a specific rule. Each number in a sequence is called a "term." The order in which terms are arranged is crucial, as each term has a specific position, often denoted as an​, where n indicates the position in the sequence.For example:2, 5, 8, 11, 14,
10 min read
Arithmetic SeriesAn arithmetic series is the sum of the terms of an arithmetic sequence, where an arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. Or we can say that an arithmetic progression can be defined as a sequence of numbers in which for every pair of
5 min read
Arithmetic SequenceAn arithmetic sequence or progression is defined as a sequence of numbers in which the difference between one term and the next term remains constant.For example: the given below sequence has a common difference of 1.1 2 3 4 5 . . . n ⇑ ⇑ ⇑ ⇑ ⇑ . . . 1st 2nd 3rd 4th 5th . . . nth TermsThe Arithmetic
8 min read
Geometric Progression or GPGeometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio.For Example, the sequence given below forms a GP with a common ratio of 2 1 2 4 8 16 . . . n⇑ ⇑ ⇑ ⇑ ⇑ . . . 1st 2nd 3rd 4th 5th . . . nt
12 min read
Geometric SeriesIn a Geometric Series, every next term is the multiplication of its Previous term by a certain constant, and depending upon the value of the constant, the Series may increase or decrease.Geometric Sequence is given as: a, ar, ar2, ar3, ar4,..... {Infinite Sequence}a, ar, ar2, ar3, ar4, ....... arn {
3 min read
Special Series in Maths - Sequences and Series | Class 11 MathsSpecial Series: A series can be defined as the sum of all the numbers of the given sequence. The sequences are finite as well as infinite. In the same way, the series can also be finite or infinite. For example, consider a sequence as 1, 3, 5, 7, … Then the series of these terms will be 1 + 3 + 5 +
10 min read
Arithmetic Progression and Geometric ProgressionArithmetic Progression and Geometric Progression: The word "sequence" in English means a collection of some numbers or objects in such a way that it has a first member, a second member, and so on. Sequences can be of anything, for example. - January, February, .... is the sequence of months in a yea
10 min read
Chapter 10: Straight Lines
Slope of a LineSlope of a Line is the measure of the steepness of a line, a surface, or a curve, whichever is the point of consideration. The slope of a Line is a fundamental concept in the stream of calculus or coordinate geometry, or we can say the slope of a line is fundamental to the complete mathematics subje
12 min read
Introduction to Two-Variable Linear Equations in Straight LinesLines are the most basic configuration in geometry. Many other geometrical shapes can be obtained from lines. Lines are referred 1-Dimensional. We can obtain higher dimensional shapes using lines. Let's understand the lines in depth. Let's say, we have two sets as follows, x = {1, 2, 3, 4, 5, 6, 7}
6 min read
Forms of Two-Variable Linear Equations - Straight Lines | Class 11 MathsLine is the simplest geometrical shape. It has no endpoints and extends in both directions till infinity. The word “straight†simply means without “bendâ€. The gradient between any two point on the line is same. Hence, we can say that, if the gradient between any two points on the line is same, then
5 min read
Point-slope Form - Straight Lines | Class 11 MathsThere are several forms to represent the equation of a straight line on the two-dimensional coordinate plane. Three major of them are point-slope form, slope-intercept form, and general or standard form. The point-slope form includes the slope of the straight line and a point on the line as the name
9 min read
X and Y InterceptThe x-intercept is the point at which the graph crosses the x-axis. At this point, the y-coordinate is zero. The y-intercept is the point at which the graph crosses the y-axis. At this point, the x-coordinate is zero. In this article, we will explore the definition of intercepts including both x and
9 min read
Slope Intercept FormThe slope-intercept formula is one of the formulas used to find the equation of a line. The slope-intercept formula of a line with slope m and y-intercept b is, y = mx + b. Here (x, y) is any point on the line. It represents a straight line that cuts both axes. Slope intercept form of the equation i
9 min read
Writing Slope-Intercept EquationsStraight-line equations, also known as "linear" equations, have simple variable expressions with no exponents and graph as straight lines. A straight-line equation is one that has only two variables: x and y, rather than variables like y2 or √x. Because it contains information about these two proper
10 min read
Graphing Slope-Intercept Equations - Straight Lines | Class 11 MathsTo graph a straight line we need at least two points that lie on the straight line. From the slope-intercept form of the given straight line, we can calculate two points on the line very easily using the information present in the equation. Consider a straight line with slope m and y-intercept c. We
7 min read
Standard Form of a Straight LineThere are several forms available to represent the equation of a straight line on the 2-dimensional coordinate plane, out of several forms three major forms are point-slope form, slope-intercept form, and general or standard form. The general or standard form is a linear equation where the degree of
11 min read