Question

How do I prove that the function f: R to R defined by f(x)= 4 + 3x is one-to-one and onto?

Answer 1

Answer:

Given:

f(x) = 3x + 4

To test one-to-one:

Let, f(x1) = f(x2)

=> 3x1 + 4 = 3x2 + 4

=> 3x1 = 3x2

=> x1 = x2

This shows that the given function f(x) is one-to-one.

Test for onto:

Let, y = f(x)

=> y = 3x + 4

=> 3x = y - 4

=> x = (y - 4)/3

For x to be real, y can take place of any real value.

Thus,

Range of the function is all real value

ie, R

Also , it is given that:

f : R –> R

Here, the codomain of the function is R.

Since, the range and the codomain of the function are same, thus the function f(x) is onto.

Hence, the function is both one-to-one and onto.(or simply bijective function)