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

To Prove: – is a bijection

Check for Injectivity:

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

⇒ f(x) = f(y)

⇒

⇒ (x – 2)(y – 3) = (x – 3)(y – 2)

⇒ xy – 3x – 2y + 6 = xy – 2x – 3y + 6

⇒ – 3x – 2y + 2x + 3y = 0

⇒ – x + y = 0

⇒ x = y

Hence, f is One – One function

Check for Surjectivity:

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

⇒ f(x) = y

⇒

⇒ x – 2 = xy – 3y

⇒ x – xy = 2 – 3y

⇒

is a real number for all y ≠ 1.

Also, for any y

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

Hence, f is onto function

Thus, Bijective function