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 Q i.e such that

⇒ f(x) = f(y)

⇒

⇒ (2x + 3)(y – 3) = (2y + 3)(x – 3)

⇒ 2xy – 6x + 3y – 9 = 2xy – 6y + 3x – 9

⇒ – 6x + 3y = – 6y + 3x

⇒ – 6x + 3y + 6y – 3x = 0

⇒ – 9x + 9y = 0

⇒ x = y

Thus, f is One – One function

Check for Surjectivity:

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

⇒ f(x) = y

⇒

⇒ 2x + 3 = y (x – 3)

⇒ 2x + 3 = xy – 3y

⇒ 2x – xy = – 3(y + 1)

⇒

Above value of x belongs to Q – [3] for y = 2

Therefore for each element in Q – [3] (co – domain), there does not exist an element in domain Q.

Hence, f is not onto function

Thus, Not Bijective function