Functions - one one and onto

1.      If f(x)= x / (1+x) and domain is set of whole numbers show that f is one- one but not onto.

2.      Let f : N to N be a function defined by f(x) = x² + x +1 , x in N then prove that f is one one and onto

3.      Let f :R to R be a function defined by f(x) = (x-m) / (x-n) where m and n are distinct. Then show that f is one –one but not onto

4.      Is the function f : N to N defined by f(n) = 2n +3 for all n in N onto?

5.      A function f is given as f : { (2,7) (3,4) (7,9) (-1,6) (0,2) (5,3)} . is this function one – one onto?

Interchange the order of the elements in the ordered pairs and form the new relation. Is this relation a function? If it is a function Is it one one onto?

6.      Show that the function f: R to R given by f(x) =cos x for all x in R is neither one-one nore onto.

7.      Let Q be the set of rational numbers. Let f: Q to Q be defined by

f(x)= 2x +3  x in Q show that f is one one. Also find a formula that defines the inverse function f^-1

8.      Let R be the set of all real numbers. Let f: R to R be f(x)= x³ – x. is this a one one mapping?

9.      Let R₀ denote the set of all non zero real numbers. Prove that the map f: R₀ to R₀ given by f(x)= 1/x where x in R₀ is both one-one and onto.

10.  Comment whether the given function is one-one, onto or one-one onto

 f: N to N by f(n) = n²+2