Question

Show that the function f: R-R defined by f(x) = cosx is neither one one nor onto

Answer 1

Step-by-step explanation:

To show---> Function f : R -------> R defined by

f ( x ) = Cosx is neither one one nor onto.

Solution---> ATQ, f: R ---------> R defined by

f ( x ) = Cosx

Let x₁ , x₂ € R

f ( x₁ ) = Cosx₁ , f ( x₂ ) = Cosx₂

Now, let,

f ( x₁ ) = f ( x₂ )

=> Cosx₁ = Cosx₂

=> Cosx₁ = Cos ( 2nπ ± x₂ )

=> x₁ = 2nπ ± x₂

So given function is not one one because more than one element of domain have same image in codomain.

Now, f( x ) = Cosx = y ( say )

x = Cos⁻¹ ( y )

We know that domain of Cos⁻¹ is [ -1 , 1 ] , so only elements of codomain which lies in domain of Cos⁻¹ have its preimage in domain of given function

For example if we take y = 2 belongs to R but it has no preimage in domain of given function .

So given function is not onto.

Answer 2

Answer:

Step-by-step explanation:

if you differentiate function you will get -sinx

which can take positive as well as negative so many one

range is subset of codomain so into

the function is many one and into

so neither one one nor onto