show that function f:N --- N, given by f(1) = f(2) = 1 and f(x) = x-1 , for - Maths - Relations and Functions

Question

show that function f:N ---> N, given by f(1) = f(2) = 1 and
f(x) = x-1 , for every x > 2 , is onto but not one to one
plz donot give the ncert soln as its not clear

Manmeet Singh · Accepted Answer

We are given f: N ---> N, given by f(1) =
f(2) = 1 and f(x) = x
â€“ 1 , for every x > 2
A function f: X â†’
Y is said to be one-one (or
injective) if the images of distinct elements of
X under f are distinct In other words, a function
f is one-one if for every x1,
x2 âˆˆ X, f
(x1) = f (x2)
implies x1 = x2
Since f(1) = f(2) = 1, but 1
â‰  2
Hence, the function is not one one
A function f: X â†’
Y is defined as onto (or
surjective) if every element of Y is the
image of some element of x in X under f
In other words, f is onto if " y âˆˆ Y, there
exist x âˆˆ X such that
f (x) = y
It can be observed that for the given function, every element of
Y has a pre-image in X
Hence the function is onto

show that function f:N ---> N, given by f(1) = f(2) = 1 and f(x) = x-1 , for every x > 2 , is onto but not one to one. plz donot give the ncert soln as its not clear.

show that function f:N ---> N, given by f(1) = f(2) = 1 and f(x) = x-1 , for every x > 2 , is onto but not one to one.

plz donot give the ncert soln as its not clear.