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.

To Prove : – Logarithmic function f : R _{+ +} → R given by f(x) = log_a x, a > 0 is a bijection.

Now, f : R₀ ⁺ → R given by f(x) = log_a x, a > 0

Check for Injectivity:

Let x,y be elements belongs to R₀ ⁺ i.e such that

So, from definition

⇒ f(x) = f(y)

⇒ log_a x = log_a y

⇒ log_a x – log_a 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

⇒ log_a x = y

⇒ x = a^y

Above value of x belongs to R₀ ⁺

Therefore, for all there exist x = a^y such that f(x) = y .

Hence, f is Onto function.

Thus, it is Bijective also