Function 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.
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)
So f(x) = x3 + 3x is an odd function.
Odd functions will have odd powers in the function.
You can find some basic types of 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).
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
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.