Q. 5 C5.0( 3 Votes )

# Classify the following functions as injection, surjection or bijection:f : N → N given by f(x) = x3

TIP: One – One Function: – A function is said to be a one – one functions or an injection if different elements of A have different images in B.

So, is One – One function

a≠b

f(a)≠f(b) for all

f(a) = f(b)

a = b for all

Onto Function: – A function is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.

So, is Surjection iff for each , there exists such that f(a) = b

Bijection Function: – A function is said to be a bijection function if it is one – one as well as onto function.

Now, f : N N given by f(x) = x3

Check for Injectivity:

Let x,y be elements belongs to N i.e such that

f(x) = f(y)

x3 = y3

x3 – y3 = 0

(x – y)(x2 + y2 + xy) = 0

As therefore x2 + y2 + xy >0

x – y = 0

x = y

Hence f is One – One function

Check for Surjectivity:

Let y be element belongs to N i.e be arbitrary, then

f(x) = y

x3 = y

not belongs to N for non–perfect cube value of y.

Since f attain only cubic number like 1,8,27….,

Therefore no non – perfect cubic values of y in N (co – domain) has a pre–image in domain N.

Hence, f is not onto function

Thus, Not Bijective also

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