We can verify whether any function is one-one by finding derivation of the function on $f\left( x \right)$ as well, If $f\left( x \right)>0$ only or $f'\left( x \right)<0$ only for the given domain then $f\left( x \right)$ is one-one, otherwise $f\left( x \right)$ will not be one-one. So, don’t confuse with the examples given in the problem, there are other examples as well. So, it can be another way as well.There are infinite examples of the functions asked in the problem. Note: Another approach to get any function to be one, we can use graphical approach, means draw the curve $y=f\left( x \right)$ as supposed in all the four option, if a line parallel to x cutting the graph at two or more points, then it will not be one-one. So, the function $f:N\to N$, given by $f\left( 1 \right)=f\left( 2 \right)=1$ is not one-one but onto. Here, y is a natural number for every ‘y’, there is a value of x which is a natural number. Let $f\left( x \right)=y$, such that $y\in N$. Since, different elements 1 and 2 have the same image ‘1’. (iv) Let the function $f:N\to N$, given by $f\left( 1 \right)=f\left( 2 \right)=1$ If b is the unique element of B assigned by the function f to the element a of A, it is written as. A is called Domain of f and B is called co-domain of f. A function f from A to B is an assignment of exactly one element of B to each element of A (A and B are non-empty sets). Mathematically, one-one is given for any function $f\left( x \right)$ as if $f\left( $ is not one-one and not onto as well for $x:R\to R$. Mathematics Classes (Injective, surjective, Bijective) of Functions. mapping of elements of range and domain are unique. If a function f is not bijective, inverse function of f cannot be defined.Hint:One-one function means every domain has distinct range i.e.For onto function, range and co-domain are equal.one to one function never assigns the same value to two different domain elements.A function is one to one if it is either strictly increasing or strictly decreasing.If f and fog are onto, then it is not necessary that g is also onto.If f and fog both are one to one function, then g is also one to one.If f and g both are onto function, then fog is also onto.If f and g both are one to one function, then fog is also one to one.A function f is decreasing if f(x) ≤ f(y) when x
It is a function which assigns to b, a unique element a such that f(a) = b. The inverse of bijection f is denoted as f -1. Inverse Functions:Bijection function are also known as invertible function because they have inverse function property.One to one correspondence function(Bijective/Invertible): A function is Bijective function if it is both one to one and onto function.It is not required that a is unique The function f may map one or more elements of A to the same element of B. Onto Function (surjective): If every element b in B has a corresponding element a in A such that f(a) = b.We can express that f is one-to-one using quantifiers as or equivalently, where the universe of discourse is the domain of the function. It never maps distinct elements of its domain to the same element of its co-domain. One to one function(Injective): A function is called one to one if for all elements a and b in A, if f(a) = f(b),then it must be the case that a = b.Mathematics | Rings, Integral domains and Fields.Mathematics | Independent Sets, Covering and Matching.
#ONTO VS ONE TO ONE GRAPHS SERIES#
Mathematics | Partial Orders and Lattices.Mathematics | Power Set and its Properties.Inclusion-Exclusion and its various Applications.Mathematics | Set Operations (Set theory).Mathematics | Introduction of Set theory.ISRO CS Syllabus for Scientist/Engineer Exam.ISRO CS Original Papers and Official Keys.GATE CS Original Papers and Official Keys.