Functions in Mathematics




Functions in MathematicsFunction is described as follows, function is a relationship between range and domain. Thinking about two sets A and B. We form the Cartesian product, we form relations. From all the relationships, we can pick a few which fulfill the rule that each component of the set A is linked to only one element of the set B. When a relation fulfills this rule, it is referred to as a function.

In this chapter, we are going to study how a function is a relation, however a relation might not be a function. Therefore, the function calculator sections assists to differentiate relation and a function. If f is a function from A → B and described as f(a) = b then a = A and b = f(a) = B is unique. A is domain and B is co domain or we can say it as Range of function f. Also, we can state that f from A to B is a relation and each relation from A to B is not a function. 

Note: f is a set of ordered sets, no two of which have the identical first coordinate.

The general notation of a function is actually y = f(x). It is also denotes as f : X ? Y in which, f is the function described between X and Y where, x is in X and y is in Y.

Characteristics of Functions

You can find several characteristics of functions, we will look at them below.

1. Odd and Even functions

A function can be odd or even.

We state a function is odd if they are symmetrical of a point, generally the origin. Such a graph when rotated through 180° about the origin (or the point of symmetry) will give the original graph.

For an odd function, f(-x)= - f(x), for all values of x in the domain.

A function is considered even if it is symmetric about the y-axis. The curve features a line of symmetry about the y-axis and will seem like a reflection of itself along the y-axis.

For an even function, f(-x) = f(x), for all values of x in the domain.  

Examples of odd and even functions:

To learn if a function f(x) is even or odd, we have to find out what occurs when it is turned to f(-x).

If f(-x) is the identical as f(x), i.e. f(-x) = f(x), then it is an even function.

And if f(-x) becomes -f(x), i.e. f(-x) = -f(x), then it is an odd function.

1. f(x) = x4 + x2 + 2

f(-x) = (-x)4 + (-x)2 + 2  =  x4 + x2 + 2

f(-x) is the same as f(x). Hence this is an even function.

A useful tip to remember is that even functions will all have even powers in the function.

2. f(x) = x3 + 3x

f(-x) = (-x)3 + 3 (-x)

= -x3 - 3x

= – (x3 + 3x)

= -f(x)

So f(x) = x3 + 3x is an odd function.

Odd functions will have odd powers in the function.

Types of Functions

You can find some basic types of functions

Surjective Functions:

Surjective function is also known as onto function. Here, every elements in the range is related to a minimum of one element in the domain. If y is element in range then there is a x such that y = f(x).

Injective Functions:

It is also known as one to one function. It fulfills the property that if f(a) = f(b) then, a = b. This is the condition with regard to injective function

Inverse Functions:

If f is a function from A to B then inverse of a function is the function defined from B to A.

It is denoted by f -1. That is, if f:A -> B then f -1 : B -> A. If f(x) = y then x = f-1(y).

Bijective Functions:

If the given function is both injective and surjective or one to one and onto then, it is known as a bijective function.



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