Question #184450

Consider the following two functions;

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

2. *g: ***RR-**defined by *f(x) *= 15x^{3}.

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

Expert's answer

- 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.

- 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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