The elementary properties of inverse trigonometric functions are valid within the principal value branches of the corresponding inverse function wherever they are defined. However, these properties are valid for a limited section of the domain of the inverse functions. Recall that, if y = sin$^{-1}$x and x= sin y then y = sin$^{-1}$x. This means … Read more

The graph of y = sin x can be visuallised in the figure below: Domain: all reals Range: [-1,1] Period: 2π Y-intercept: (0,0) When you restrict the domain of sin x to the interval –π/2 ≤ x ≤ π/2, the following properties should hold: 1.y = sin x is an increasing function. 2.y = sin … Read more

Recall from previous chapters that inverse of a function f can be written as f-1, where the expression sin-1 is read as “the inverse sine.” In this notation,-1 indicates the inverse and not the reciprocal of the sine function.Other names for inverse trigonometric functions are arcsine, arccosine, and arctangent functions. The inverses are not functions … Read more

one-to-one function one-to-one function or injective function is one of the most common functions used.  One-to-One functions define that each element of one set say Set (A) is mapped with a unique element of another set, say, Set (B). To understand this, let us consider ‘f’ is a function whose domain is set A. The … Read more

If we combine two functions in such a way that the output of one function becomes the input to another function, then this is called as composite function. Consider three sets X, Y and Z and let f: X → Y and g: Y → Z. As per this, under f, an element x∈ X is mapped to … Read more

Let f: X → Y. Now, let f represent a one to one function and y be any element of Y, there exists a unique element x ∈ X such that y = f(x).Then the map f−1: f[X] →X That associates to each element is called as the inverse map of f. The function f(x) = x5 and g(x) = x1/2 have the following property: f(g(x)) = f(x1/5) = (x1/5)5 = x g(f(x)) = g(x5) = (x5)1/5 = x … Read more

*, a binary operation on a set A is basically a function *: A × A → A. We denote * (a, b) by a * b. if a * b = b * a for every a, b ∈ X, a binary operation * on the set X is called commutative. A binary operation … Read more

A symmetric relation is nothing but a type of a binary relation. The relation “is equal to”, is symmetric because if a = b is true then b = a is also true. In formal terms, a binary relation Rover a set X is symmetric only in the following condition: If RT represents the converse of R, then R is symmetric if and only if R = RT Let A be a set and R be the relation … Read more

A relation which is reflexive, symmetric and transitive is an Equivalence relation on set.Relation R, defined in a set A, is said to be an equivalence relation only on the following conditions: (i) aRa for all a ∈ A, that is,R is reflexive. (ii) aRb⇒bRa for all a, b ∈ AR, that is, is symmetric … Read more

In maths, any relation R over a set X is called reflexive if every element of X is related to itself. In formal terms, this may be written as ∀x ∈ X : x R x. The relation “is equal to” on the set of real numbers is an example of a reflexive, since every real number is equal to itself. A reflexive relation is said to possess reflexivity … Read more