Question

The function is $ f(x)=\frac{1}{2-\cos \,3x},\,x\,\in \left[ 0,\frac{\pi }{3} \right], $ is

• A. one-one, but not onto
• B. onto, but not one-one
• C. one-one as well as onto
• D. neither one-one nor onto

Answer 1

The Correct Option is C

Given, $ f(x)=\frac{1}{2-\cos \,3x},\,x\,\in \left[ 0,\frac{\pi }{3} \right] $ For one - one Let $ f({{x}_{1}})=f({{x}_{2}}) $
$ \Rightarrow $ $ \frac{1}{2-\cos \,3{{x}_{1}}}=\frac{1}{2-\cos \,3\,{{x}_{2}}} $
$ \Rightarrow $ $ 2-\cos \,3{{x}_{1}}=2-\cos \,3{{x}_{2}} $
$ \Rightarrow $ $ \cos \,3{{x}_{1}}=\cos \,3{{x}_{2}}\,\,\Rightarrow \,\,{{x}_{1}}={{x}_{2}} $
$ \Rightarrow $ f is one-one For onto Let $ y=f(x),\,\,y\,\,\in $ codomain
$ \Rightarrow $ $ y=\frac{1}{2-\cos \,3x} $
$ \Rightarrow $ $ y(2-\cos \,3x)=1 $
$ \Rightarrow $ $ 2-\cos \,3x=\frac{1}{y} $
$ \Rightarrow $ $ \cos \,3x=2-\frac{1}{y} $
$ \Rightarrow $ $ x=\frac{1}{3}\,{{\cos }^{-1}}\left( 2-\frac{1}{y} \right) $
Here, for all $ y\,\in $ codomain there exist $ x\,\in $ domain. so $ f(x) $ is onto.
Here for all $ y\,E $ codomain there exist $ x\,\in $ domain. so is on to. $ f(x) $

Answer 2

Ans. Surjective Function is another name for the onto function. A function is described as a relationship between values or elements of one set and values or elements of another set such that the elements of the second set may be defined equally well by the components of the first set. A function consists of distinct kinds, which often specify the connection between two sets that follow different patterns. Onto function, many-to-one function, one-to-one function, etc. are some examples of the various types of function.

The function is a process that relates elements of two different sets. One set is called the domain while the other is referred to as the codomain. Functions can be explained in simple words as ‘for every set of inputs, there is a unique set of outputs’. It is important to note that the output and input sets should have some elements for a function to exist, which means it must not be empty sets. 

A function is often represented as f (x) = y. There are many distinct types of functions, including equivalence relations, periodic functions, many-to-one relations, algebraic relations, onto relations, into relations, rational relations, one-to-one relations, linear, quadratic, cubic, even, and odd relations. 

There is just one and only one element in the codomain for each element of a function in the domain. There can be no relationship between any two codomain elements and any element in the domain. It is possible to correlate the same element in the codomain with two different items in the domain.

The range denotes the actual input of the function, the codomain states possible outcomes, and the domain is basically what can go into the function.

• All elements in the onto function have a right inverse.
• All function is said to be Surjective if it is a right inverse.
• It results in onto function only if we compose onto functions.