The Linear Nature of the Function ( f(x) ) and Its Application

In this article, we will explore the nature of the function ( f(x) ) when it satisfies the given functional equation:

Functional Equation

Given the functional relation:

( f(x) cdot f(y) f(2xy) - 3f(xy) 6x ) (1)

When we interchange the roles of ( x ) and ( y ), we get the following:

( f(x) cdot f(y) f(2xy) - 3f(xy) 6y ) (2)

By comparing (1) and (2), we derive:

( -3f(x) 6x -3f(y) 6y )

This simplifies to:

( f(x) - f(y) 2x - 2y )

Letting ( y 0 ), we get:

( f(x) 2x C ) where ( C f(0) ).

In other words, ( f ) is a linear function!

Determining the Value of ( C )

To determine the value of ( C ), we substitute ( f(x) 2x C ) into the original functional equation (1):

( (2x C) cdot (2y C) (2xy)^2 - 3xy 6x )

This simplifies to:

( 2xy C(2x 2y 2) C^2 6xy - 3xy 6x )

Further simplification gives us:

( 2Cxy C^2 6xy - 3xy 6x )

Pulling out the common term ( xy ) from both sides, we get:

( 2Cx C^2 3xy - 3xy 6x )

This reduces to:

( 2Cx C^2 6x )

Next, we separate the terms involving ( C ) and ( x ):

( 2C 6 ) and ( C^2 6C )

Solving these, we find:

( C 3 )

Thus:

( f(x) 2x 3 )

Application to Find ( f(1009) )

Using the discovered linear relationship, we can now determine the value of ( f(1009) ):

( f(1009) 2 cdot 1009 3 2018 3 2021 )

Therefore, ( f(1009) 2021 )!