*a function can be inversed if it is bijective....*

*a function is bijective if it is one-one(injective) and onto(surjective)...*

*ALGORITHM to check one- one*

*1) take two arbitrary elements x,y (say) in the domain of f...*

*2)put f(x)= f(y)*

*3) solve f(x)= f(y) gives x= y only..then f: A---->B is one- one*

*NOW...lets prove question is one- one*

*now let x,y be arbitrary elements in the domain of f...such that*

* f(x) = f(y)*

* x*^{3 }= y^{3 }

^{ }x = y

*hence ..it is one- one*

*now ALGORITHM for ONTO(surjective)*

*let f:A--->B*

*1) choose any arbitrary elements y in B.*

*2) put f(x) = y *

*3) solve the equation f(x) = y for x and obtain x in terms of y..let x= h(y)*

*4)if for all values of y∈B, the values of x obtained from x = h(y) are in A, then f is onto..if there are some y∈B for which x, given by x= g(y) is not in A..then f is not onto..*

*now prove above question is ONTO(surjective)..*

*f(x) = y*

*x*^{3 }= y

*x= (y)*^{1/3}

*clearly , for all y∈ R, **(y)*^{1/3 }is a real number

*thus, for all y∈ R, B(co-domain) there exists x= (y)*^{1/3 }in R, A(domain) such that

*f(x) = x*^{3 }= y

*hence, f:R-->R is onto function..and hence above function is bijective*

*NOW , find inverse*

*ALGORITHM for INVERSE*

*let f: A-->B be a bijection ..*

*1) put f(x) = y *

*2) solve the equation f(x) = y for x and obtain x in terms of y*

*3) the relation obtained in above step-2 replace x by f*^{-1}(y) to obtain the requred inverse of y..

*4)later convert all in terms of x..by just changing y to x.. *

*now lets solvee questtion..*

*f(x) =y*

*x*^{3 }= y

*x =(y)*^{1/3 }put in place of x put it as * f*^{-1}(y)

* f*^{-1}(y) =(y)^{1/3}

*now we can write it as: by changing y to x*

* f*^{-1}(x) =(x)^{1/3}

*now..we can clearly see..that*** f**^{-1}**(x) is equal to g(x) **and hence f(x) is inverse of g(x)

*now lets prove g(x) is inverse of *

*g(x) =y*

*y =(x)*^{1/3}

*x= y*^{3 }

*g*^{-1}(y) = y^{3} we can write it as

*g*^{-1}(x) = x^{3} now we can clearly see **that g**^{-1}(x) is equal to f(x)..g(x) is inverse of f(x)