Prove that the function `f: R rarr R` given by `f(x)=2 x` is one-one and onto.

Show that the function f:R rarr R given by f(x)=x^3 is injective.

Show that the function f : R_** rarr R_** defined by f (x) = 1/x is one-one and onto,where R_** is the set of all non-zero real numbers. Is the result true, if the domain R_** is replaced by N with co-domain being same as R_** ?

Prove that the function f: N rarr N , defined by f(x)=x^2+x+1 is one-one but not onto.

Show that the function f:RtoR defined by f(x)=2x-3 is one-one and onto. Find f^-1

Show that the modulus function f: R rarr R given by f(x) = |x| , is neither one-one nor onto.

Prove that the greatest integer function f : R rarrR given by f(x) =[x] is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Prove that the function f: Q rarr Q given by f(x)=2 x-3 for all x in Q (the set of all rational numbers) is a bijection.

Let A = R - {-3} and B = R - {1} Consider the function f : A rarr B defined by f(x) = (x-2)/(x-3) . Is f one-one and onto ? Justify your answer.