What is a Function in Math?
A Function from set M to set N is a binary relation or a rule which links or plots or pictures each and every component of set M with a component in set N. The purpose of this chapter is to make you learn about various types of functions so that you can become acquainted with the types. You will also come to know that each type has its own individual graphs. Examples of the different types of functions are shown below.
The denotation of function in Mathematics
A function from set M to set N is denoted by:
F: M→N
We chiefly use F, G, H to denote a function
We can also denote a Mathematical class of any function using the following method:
Tabulation Method
Graph method
Arrow Diagram method
A function is defined as a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A and B be two non-empty sets, mapping from A to B will be a function only when every element in set A has one end and only one image in set B. It can be defined that a function is a special relation which maps each element of set A with one and only one element of set B. Both sets A and B must not be empty. A function will define a particular output for a particular input. Therefore, f: A → B is a function such that for a ∈ A there is a unique element b ∈ B such that (a, b) ∈ f. For every Mathematical expression if it has an input value and a resulting answer can be presented as a function.
Types of Functions in Mathematics with Examples
Types of functions are generally classified into four different types: Based on Elements, Based on Equation, Based on Range, and Based on Domain.
1. Based on Elements:
One One Function
Many One Function
Onto Function
One One and Onto Function
Into Function
Constant Function
2. Based on Equation:
Identity Function
Linear Function
Quadratic Function
Cubic Function
Polynomial Functions
3. Based on the Range:
Modulus Function
Rational Function
Signum Function
Even and Odd Functions
Periodic Functions
Greatest Integer Function
Inverse Function
Composite Functions
4. Based on the Domain:
Algebraic Functions
Logarithmic Functions
Types of Function - Based on Elements
1. One-To-One Function.
A Mathematical function is said to be a One-To-One Function if every component of the Domain function possesses its own and unique component in Range of the Function. That being said, a function from set M to set N is considered a One-To-One Function if no two or more elements of set M have the same components mapped or imaged in set N. Also, that no two or more components refined through the function provide the similar output.
For Example:
When f: M→N is described by formula y= f (x) = x³, the function “f” is stated to be a One-To-One function since a cube of different numbers is always different itself.
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2. Onto Function.
A Function is Onto Function if two or more components in its Domain have the same component in its Range.
For Example:
If set M= {M, N, O} and set N= {1,2}
And “f” is a function by which f: M→N is described by:
Then the function “f” is regarded as Onto Function.
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3. Into Functions
A function is said to be an Into function in which there is an element of co-domain Y and does not have a pre-image in domain X.
Example:
Take into account, P = {P, Q, R}
Q = {1, 2, 3, 4} and f: P→ Q in a way
f = {P,1, Q,2, R,3}
In the function f, the range i.e., {1, 2, 3} ≠ co-domain of Y i.e., {1, 2, 3, 4}
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4. One - One Into Functions
The function f is said to be one-one into a function if there exists different components of X and have distinctive unique images of Y.
Example: Prove one-one into function from below set
X = P,Q,R
Y = [1, 2, 3, and 4} and f: X → Y in a way
f = {P,1, (Q,3, R,4}
X = P,Q,R
Y = [1, 2, 3, and 4} and f: X → Y in a way
f = {P,1, (Q,3, R,4}
Thus, function f is a one-one into function
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5. Many-One Functions
The function f is many-one functions if two or more different elements in X have the same image in Y.
Example: Prove many-one function
Taken, X = 1,2,3,4,5
Y = XYZ
X,Y,Z and f: X → Y
Thus and thus f = {1,X,2,X,3,X,4,Y,5,Z}
Hence, function f is a many-one function
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6. Many-One Into Functions
The function f is a many-one function only if it is—both many ones and into a function.
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7. Many-One Onto Functions
The function f is many-one onto function only if is –both many ones and onto.
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8. Constant Function
A constant function is one of the important forms of a many to one function. In this domain every element has a single image.
The constant function is in the form of
f(x) = K, where K is a real number.
Types of Function - Based on Equation
Identify Function: The function that has the same domain and range.
Constant Function: The polynomial function of degree zero.
Linear Function: The polynomial function of degree one.
Quadratic Function: The polynomial function of degree two.
Cubic Function: The polynomial function of degree three.
Types of Functions - Based on Range
Modulus Function
The modulus function is the type of function that gives the absolute value of the function, irrespective of the sign of the input domain value.
The modulus function is defined as f(x) = |x|.
The input value of 'x' can be a negative or a positive expression.
Rational Function
A Rational Function is the type of function that is composed of two functions and expressed in the form of a fraction X.
A rational function is of the form
f(x)/g(x), and g(x) ≠ 0.
Signum Function
The signum function is the type of function that helps to know the sign of the function and does not give the numeric value or any other values for the range.
Even and Odd Function
The even and odd function are the type of functions that are based on the relationship between the input and the output values of the function.
Periodic Function
The function is said to be a periodic function if the same range appears for different domain values and in a sequential manner.
Inverse Function
The inverse of a function is the type of function in which the domain and range of the given function is reverted as the range and domain of the inverse function.
The inverse function f(x) is denoted by f-1(x).
Greatest Integer Function
The greatest integer function is the type of function that rounds up the number to the nearest integer less than or equal to the given number.
The greatest integer function is represented as
f(x) = ⌊x⌋.
Composite Function
The composite function is the type of function that is made of two functions that have the range of one function forming the domain for another function.
Types of Functions - Based on Domain
Algebraic Function
An algebraic function is the type of function that is helpful to define the various operations of algebra. This function has a variable, coefficient, constant term, and various arithmetic operators such as addition, subtraction, multiplication, division.
Trigonometric Functions
The trigonometric function is the type of function that has a domain and range similar to any other function. The 6 trigonometric functions are :
f(θ) = sinθ, f(θ) = tanθ, f(θ) = cosθ, f(θ) = secθ, f(θ) = cosecθ.
Logarithmic Functions
Logarithmic functions are the type of function that is derived from the exponential functions. The logarithmic functions are considered to be the inverse of exponential functions.
Solved Example of Functions
1. Find the inverse function of the function f(x) = 5x + 4.
Solution: The given function is f(x) = 6x + 4
It is rewritten as y = 6x + 4 and then simplified to find the value of x.
y = 6x + 4
y - 4 = 6x
x = (y - 4)/6
f-1(x) = (x - 4)/6
Ans: So the answer of this inverse function is f-1(x) = (x - 4)/6
2. For the given functions f(x) = 3x + 2 and g(x) = 2x - 1, find the value of fog(x).
Solution: The given two functions are f(x) = 3x + 2 and g(x) = 2x - 1.
The function fog(x) is to be found.
fog(x) = f(g(x))
= f(2x-1)
= 3(2x - 1) + 2
= 6x - 3 + 2
= 6x - 1
Ans: Therefore fog(x) = 6x - 1
Representation of Functions
The functions can be represented in three ways: Venn diagrams, graphical formats, and roster forms.
Venn Diagram: The Venn diagram is one of the important formats for representing the function. The Venn diagrams are generally presented as two circles with arrows connecting the element in each of the circles.
Graphical Form: It is said that every function is easy to understand if they are represented in the graphical form with the help of the coordinate axes. The function in graphical form, helps to understand the changing behavior of the functions if the function is increasing or decreasing.
Roster Form: Roster form is a set of a simple Mathematical representation of the set in Mathematical form. The domain and range of the function in Roster form are represented in flower brackets with the first element of a pair representing the domain and the second element representing the range.
Practice Problems
Practice Problem-1:
Alex leaves his apartment at 5:50 a.m. and goes for a 9-mile jog. He returns at 7:08 a.m. to answer the following questions, assuming Alex runs at a persistent pace.
Report the distance D (in miles) Alex jogs as a linear function of his run time ‘t’ (in minutes).
Draw a graph of D
Simplify the sense of the slope.
Solution-1:
(i) At time t=0, Alex is at his apartment, thus, D (0) =0
At time t= 78 minutes, Alex completed running 9 mi, thus, D (78) =9.
The slope of the linear function comes about as:-
m=9−0 / 78−0= 3 / 26
The y-intercept is (0, 0), thus, the linear equation for this function is
D (t) =3/26 t
(ii) Now, to graph D, execute the fact that the graph cross over the origin and has slope m=3/26
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(iii) The slope m= 3/26 ≈ 0.115 reports the distance (in miles) Alex runs per minute or his average velocity.
Fun Facts
As per Math processing, there are an infinite number of functions, much more than what you learned in this chapter
Different Mathematical functions can make us protected in life as being misemployed, deceived or exploited.
FAQs on Types of Functions
1. What is meant by a ‘function’ in mathematics, and how is it different from a general relation?
A function is a specific type of relation between two sets where every element of the first set (domain) is paired with exactly one element of the second set (codomain). Unlike general relations, a function ensures one unique output for each input, which is essential for representing mathematical models and real-world scenarios.
2. How are functions classified based on their elements, and what are some examples?
Functions are classified based on the mapping of elements between sets:
- One-one (Injective) Function: Each element of the domain maps to a unique element in the codomain, e.g., f(x) = x + 1.
- Onto (Surjective) Function: Every element of the codomain has at least one pre-image in the domain.
- Many-one Function: Multiple elements of the domain map to the same codomain element, e.g., f(x)=x2 for x = 1 and x = -1 both mapping to 1.
- Into Function: Not all elements of the codomain are mapped from the domain.
3. What are the major types of functions based on algebraic expressions?
Common types based on their algebraic form include:
- Constant Function: f(x) = k, where output is always a fixed value.
- Linear Function: f(x) = ax + b; produces straight-line graphs.
- Quadratic Function: f(x) = ax2 + bx + c; forms a parabola.
- Cubic & Polynomial Functions: Involve variables with degree three or higher.
4. What is the significance of the modulus, signum, and greatest integer functions, and how do their outputs differ?
Modulus Function gives the absolute value regardless of sign, e.g., |x|.
Signum Function indicates the sign of 'x' as -1, 0, or 1 depending on negative, zero, or positive input.
Greatest Integer Function (also called floor function) gives the largest integer less than or equal to x, represented as ⌊x⌋. Each of these functions helps model scenarios like distance (modulus), direction (signum), and rounding numbers (greatest integer).
5. How is an inverse function defined, and what conditions must a function satisfy to have an inverse?
An inverse function reverses the mapping of the original function. A function must be both one-one (injective) and onto (surjective) (i.e., bijective) for its inverse to exist. For example, if f(x) = 2x + 3 (a bijective function on real numbers), its inverse is f-1(x) = (x - 3)/2.
6. In what ways can functions be visually represented, and why is graphical representation important for understanding types of functions?
Functions can be represented by Venn diagrams (showing associations), roster form (listing pairs), and graphs (plotting on coordinate axes). Graphical representation is crucial as it makes it easy to identify function properties such as domain, range, behavior (increasing/decreasing), and types like linear or quadratic at a glance.
7. What are some real-life applications of different types of functions?
Functions model various real-life phenomena:
- Linear functions describe speed-time graphs, or distance as a function of time.
- Quadratic functions represent projectile motion and area problems.
- Exponential and logarithmic functions are used in population studies, radioactive decay, and financial calculations.
- Trigonometric functions are essential in engineering and wave motion.
8. How does a composite function work, and what does fog(x) mean?
A composite function combines two functions such that the output of one becomes the input of the other. If f(x) = 3x + 2 and g(x) = 2x - 1, then fog(x) = f(g(x)) = 3(2x-1) + 2 = 6x - 1. This allows more complex mappings and is widely used in mathematics and real-world processes.
9. How can misconceptions about domain and range lead to errors in identifying function types?
Misconceptions regarding domain and range can cause a function to be incorrectly classified. For example, assuming all real numbers are part of a function's domain without checking restrictions (such as denominator not zero in rational functions) can lead to errors. Clearly identifying domain and range is necessary for correct function analysis.
10. If two different types of functions are combined, such as a linear and periodic function, what kind of graph and properties can result?
Combining a linear and a periodic function (e.g., f(x) = x + sin x) results in a graph that shows oscillation (from the periodic part) superimposed on a straight-line increase (from the linear part). This combination is common in modeling situations where a steady trend is influenced by repeating fluctuations, such as temperature changes over time.
