Table of Contents
Introduction
A function is a relationship between two sets in which each element in the first set corresponds to exactly one element in the second set. This relationship can be represented using various mathematical notations, such as f(x) or y = f(x).
Cartesian Product of Sets
Given two sets A and B, the set of all ordered pairs (a, b) such that a ∊ A and b ∊ B is called the Cartesian product of A and B, and is denoted by A × B It is read “A cross B”.
In the set-builder notation, we have
A×B = {(a,b) : a ∊ A, b ∊ B}
Exercise – 2.1
Introduction
A function is a way to describe how one set of things relates to another set of things. It establishes a connection or relationship between the elements of the first set and the elements of the second set. In this relationship, each element from the first set corresponds to exactly one element in the second set, and no element is left unmatched.
We can represent this relationship using mathematical notations. For example, f(x) indicates that the function is named “f” and it takes an input value “x” from the first set, producing a corresponding output value. Alternatively, we can use the equation form y = f(x), where the function f(x) determines the value of “y” based on the input value “x”. These notations help us express and work with functions in a concise and meaningful way.
Inverse function
An inverse function, also known as an anti-function, is a function that can undo or reverse the action of another function. In simpler terms, if we have a function “f” that takes an input “x” and produces an output “y,” then the inverse of “f” will take “y” as input and produce “x” as output.
To represent the inverse function, if the original function is denoted by ‘f’ or ‘F’, the inverse function is denoted by ‘f⁻¹’ or ‘F⁻¹’. It’s important not to confuse the ‘-1’ notation used for the inverse function with exponents or reciprocals. In this context, the ‘-1’ signifies the inverse relationship between the original function and its inverse.
Exercise – 2.2
Introduction of types of Algebraic Functions
Functions, that can be formed from a real variable x with the help of some algebraic operations such as addition, subtraction, multiplication, division and extraction of roots, are called the algebraic functions. Some special types of algebraic functions are defined below:
a. The Identity Function
Let A be any set. The function f:A→A defined by, y = f(x) = x for x ∊ A is called the identity function. It is usually denoted by Iₐ.
b. The Constant Function
Let A be any set and B = {c}. Then, the function f : A→B defined by y = f(x) = c for x ∊ A is called the constant function. In other words, a function is said to be a constant function if all its functional values are the same. (i.e if the range of the function is a singleton set).
If A = R the set of real numbers and c is a real number, the graph of y = f(x) = c is a straight line parallel to the x-axis at a distance of c units from the x-axis:
c. The Linear Function
Let A and B be any two sets. Then, a function f: A→B defined by y = f(x) = mx + c for x ∊ A, where m and c are constants, is called a linear function.
If A=B=R, the set of real numbers, the function defined by y= f(x)=x+1 is a linear function. Its graph is a straight as shown in figure given aside.
