Check the injectivity and surjectivity of the following functions:(i) `f : N ->N`given by `f(x)=x^2`(ii) `f : Z-> Z`given by `f(x)=x^2`(iii) `f : R ->R`given by `f(x)=x^2`(iv) `f : N-> N`given by `f(x)=x^3`(v) `f : Z ->
Lost your password? Please enter your email address. You will receive a link and will create a new password via email.
Please briefly explain why you feel this question should be reported.
Please briefly explain why you feel this answer should be reported.
Please briefly explain why you feel this user should be reported.
(i) `f:N to N and f(x) = x^(2)`
Let `x, y in N and f(x) -f(y)`
`rArr ” ” x(2) =y^(2)`
`rArr ” ” x=y ( because x,y in N)`
`therefore f` is one-one.
Let `f(x) = y` where `y in N`
`rArr x ^(2) =y `
`rArr x = pm sqrty notin N ` if `y =2`
`therefore f` in not onto.
Therefore, `f` is one-one but not onto.
(ii) `f: Z to Z and f(x) = x^(2)`
Let `x, y in Z and f(x) = f(y)`
`rArr x ^(2) = y^(2) rArr x = pm y `
`therefore f `is not one-one.
Again, let `f(x) = y` where `y in Z`
`rArr x ^(2) =y`
`rArr x = pm sqrty notin Z ` If ` y = 2 `
` therefore f` is not onto.
Therefore, `f` is neither one-one nor onto.
(iii) `f : R to R and f(x) = x^(2)`
Let `x, y in R and f(x) = f(y)`
`rArr x^(2) = y^(2) rArr x = pm y `
`therefore f ` is not one-one
Again, let `f(x) = y ` where `y in R`
`rArr x^(2) = y`
`rArr x = pm sqrty notin R` if ` y = -2`
` therefore f` is not onto.
Therefore, `f` is neither one-one nor onto.
(iv) `f: N to N and f(x) = x^(3)`
Let `x , y in N and f(x) = f(y)`
`rArr ” “x ^(3) = y^(3) rArr x =y `
`therefore f` is one-one.
Let `” ” f(x) =y ” ” ` where `y in N `
`rArr x ^(3) = y `
` rArr ” “x = y ^(1//3) notin N ` if ` y = -2 `
`therefore f ` is not onto.
Therefore, `f` is one-one but not onto.
(v) `f: Z toZ and f(x) = x^(3)`
Let ` x , y in Z and f(x) = f(y)`
`rArr x ^(3) = y^(3) rArr x =y`
`therefore f` is one-one.
Again let `f(x) -y ` where `y in Z`
`rArr x ^(3) = y`
`rArr x = y^(1//3) notin Z ` if `y= 2`
`therefore f ` is not onto.
Therefore, `f` is one-one but not onto.