Q. 5 Q

# Classify the foll

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: R R given by

Check for Injectivity:

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

f(x) = f(y)

xy2 + x = yx2 + y

xy2 + x – yx2 – y = 0

xy (y – x) + (x – y) = 0

(x – y)(1 – xy) = 0

Case i :

x – y = 0

x = y

f is One – One function

Case ii :

1 – xy = 0

xy = 1

Thus from case i and case ii f is One – One function

Check for Surjectivity:

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

f(x) = y

x = x2y + y

x – x2y = y

Above value of x belongs to R

Therefore for each element in R (co – domain) there exists an element in domain R.

Hence, f is onto function

Thus, Bijective function

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