Showing Bijective Functions Examples

In this class, We discuss Showing Bijective Functions Examples.

The reader should have prior knowledge of one-to-one and onto functions. Click Here.

Example 1:

Show that F: R -> R defined by f(x) = 2x + 1 is a bijective function.

To show the given function is one-to-one.

Assume F(x) = F(y) and show x = y.

If x is not equal to y, it is not a one-to-one function.

f(x) = f(y)

2x + 1 = 2y + 1

2x = 2y

x = y

The function is one-to-one.

Showing onto function:

f(x) = y

2x + 1 = y

x = (y -1)/2

(y – 1) /2 ∈ R

Every element in the codomain has a pre-image in the domain.

Example 2:

f: z+ -> z+ f(x) = 3x. is it bijective?

One to one:

f(x) = f(y)

3x = 3y

x = y

Onto function:

Five is in the codomain.

f(x) = y

3x = 5

x = 5/3

5/3 is not in z+

the function is not onto function.

Example 3:

f: R -> R f(x) = x^2 is it bijective?

One-to-one function:

f(x) = f(y)

x^2 = y^2

We can not cancel squares and make x = y.

so the function is not one-to-one.

Onto function:

y = -1

f(x) = y

x^2 = -1

x = +- i

x is an imaginary number.

We do not have a pre-image for negative numbers—so it is not an onto function.