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FUNCTİONS
Types of functions
Even Function
Definition. A function f is even if the following equation holds for all x and −x in the domain of f : f(x)=f(−x) f ( x ) = f ( − x ) Geometrically, the graph of an even function is symmetric with respect to the y -axis, meaning that its graph remains unchanged after reflection about the y -axis.
Odd Function
A function is odd if −f(x) = f(−x), for all x. The graph of an odd function will be symmetrical about the origin. For example, f(x) = x3 is odd. That is, the function on one side of x-axis is sign inverted with respect to the other side or graphically, symmetric about the origin.
Into Fuction
Into function is a function in which the set y has atleast one element which is not associated with any element of set x. Let A={1,2,3} and B={1,4,9,16}. Then, f:A→B:y=f(x)=x2 is an into function, since range (f)={1,4,9}⊂B.
Onto Function
Onto function could be explained by considering two sets, Set A and Set B, which consist of elements. If for every element of B, there is at least one or more than one element matching with A, then the function is said to be onto function or surjective function.
Unit Function
In number theory, the unit function is a completely multiplicative function on the positive integers defined as: It is called the unit function because it is the identity element for Dirichlet convolution. It may be described as the "indicator function of 1" within the set of positive integers.
Linear Function
In Mathematics, a linear function is defined as a function that has either one or two variables without exponents. It is a function that graphs to the straight line.
Constant Function
A constant function is a linear function for which the range does not change no matter which member of the domain is used. f(x1)=f(x2) for any x1 and x2 in the domain.
With a constant function, for any two points in the interval, a change in x results in a zero change in f(x) .
Graph Function
On to One Function
A function f is 1 -to- 1 if no two elements in the domain of f correspond to the same element in the range of f . In other words, each x in the domain has exactly one image in the range. And, no y in the range is the image of more than one x in the domain.
Defination of function
n mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The modern definition of function was first given in 1837 by the German mathematician Peter Dirichlet:
History of the function
The mathematical concept of a function emerged in the 17th century in connection with the development of the calculus; for example, the slope {\displaystyle \operatorname {d} !y/\operatorname {d} !x}{\displaystyle \operatorname {d} !y/\operatorname {d} !x} of a graph at a point was regarded as a function of the x-coordinate of the point. Functions were not explicitly considered in antiquity, but some precursors of the concept can perhaps be seen in the work of medieval philosophers and mathematicians such as Oresme.
Mathematicians of the 18th century typically regarded a function as being defined by an analytic expression. In the 19th century, the demands of the rigorous development of analysis by Weierstrass and others, the reformulation of geometry in terms of analysis, and the invention of set theory by Cantor, eventually led to the much more general modern concept of a function as a single-valued mapping from one set to another.
