Answer to Question #184450 in Linear Algebra for Patrick Buthelezi

Question #184450

Consider the following two functions;

1. f: R-R defined by f(x) = 4x-15.

2. g: RR-defined by f(x) = 15x3.

Prove that both f and g are one-to-one correspondence.

1
Expert's answer
2021-05-07T09:04:01-0400
  1. Given defined by

Require to prove that is a one-to-one correspondence.

To prove that is a one-to-one correspondence, we need to prove that is one-one and onto.


One - one:

Let such that

We prove that

Now

So, we proved that for all

Therefore, is one - one.


Onto:

Let (codomain)

We need to find (domain) such that

Now

So, we proved that for each (codomain) such that

Therefore, is onto.

is one - one and onto implies is one - to - one correspondence.

Hence, is a one-to-one correspondence.

  1. Given: defined by

Require to prove that is a one - to - one correspondence.


One - one:


Let such that

Now

We proved that for all

Therefore, is one - one.


Onto:


Let (codomain)

We require to find (domain) such that

Now

So, we proved that for each such that

Therefore, is onto.


Hence, is a one - to one correspondence.


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