Give an example of a function which is

(i) one - one but not onto

(ii) one - one and onto

(iii) neither one - one nor onto

(iv) onto but not one - one.

(i) one - one but not onto

f(x) = 6x

For One - One

f(x₁) = 6x₁

f(x₂) = 6x₂

put f(x₁) = f(x₂) we get

6x₁ = 6x₂

Hence, if f(x₁) = f(x₂) , x_{1 =} x₂

Function f is one - one

For Onto

f(x) = 6x

let f(x) = y ,such that y∈N

6x = y

⇒

If y = 1

x =

which is not possible as x∈N

Hence, f is not onto.

(ii) one - one and onto

f(x) = x⁵

⇒y = x⁵

Since the lines do not cut the curve in 2 equal valued points of y, therefore, the function f(x) is one - one.

The range of f(x) = ( - ∞,∞) = R(Codomain)

∴f(x) is onto

∴f(x) is one - one and onto.

(iii) neither one - one nor onto

f(x) = x²

for one one:

f(x₁) = (x₁)²

f(x₂) = (x₂)²

f(x₁) = f(x₂)

⇒(x₁)² = (x₂)²

⇒x₁ = x₂ or x₁ = - x₂

Since x₁ does not have a unique image it is not one - one

For onto

f(x) = y

such that y∈R

x² = y

⇒x =

If y is negative under root of a negative number is not real

Hence,f(x) is not onto.

∴f(x) is neither onto nor one - one

(iv) onto but not one - one.

Consider a function f:Z→N such that f(x) = |x|.

Since the Z maps to every single element in N twice, this function is onto but not one - one.

Z - integers

N - natural numbers.