**f**and

**g**that map positive integers to positive integers. It is known that

**f(f(f(x)))=g(g(x))**for all positive integers

**x**.

It is given that

**g(x)=ax+b**.

Jasmine and Luke and perfect logicians and functional experts.

Jasmine is told the value of

**a+b**. Luke is given

**a**.

Jasmine: If you tell me

**a**, I would know infinitely many values of

**f(x)**.

Luke: If I told you

**a**, and you were given a random positive integer

**y**, could you determine

**f(y)**.

Jasmine: It depends.

Luke: Then, if I were given any positive integer

**y**, I could determine

**f(y)**.

Jasmine: Then so could I.

Jasmine is then given

**y=2019**.

What value of

**f(2019)**does she give?