Table of Contents
Functions
A function is a set of instructions that performs a specific task. Functions are written in a programming language and can be used in a variety of applications. Functions can be used to create a custom toolbar, add features to a program, or perform complex mathematical operations.
Functions are often used in conjunction with variables. A variable is a value that can change, while a function is a set of instructions that always produces the same result. For example, a function might calculate the square root of a number, while the variable might store the number to be square rooted.
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:
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One One Function
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Many One Function
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Onto Function
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One One and Onto Function
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Into Function
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Constant Function
2. Based on Equation:
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Identity Function
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Linear Function
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Quadratic Function
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Cubic Function
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Polynomial Functions
3. Based on the Range:
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Modulus Function
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Rational Function
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Signum Function
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Even and Odd Functions
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Periodic Functions
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Greatest Integer Function
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Inverse Function
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Composite Functions
4. Based on the Domain:
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Algebraic Functions
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trignometric function
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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.
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
