Solving for ( f(x) ) Given ( f(f(x)) 3x 2 )

Functional equations can be quite intriguing, and solving them involves a blend of algebraic manipulation and logical reasoning. In this article, we will explore how to find the function ( f(x) ) given the functional equation ( f(f(x)) 3x 2 ).

Understanding the Functional Equation

A functional equation is an equation in which the unknown is a function. In the given problem, we are given:

[ f(f(x)) 3x 2 ]

We are to find the form of ( f(x) ).

Assuming a Linear Form for ( f(x) )

Let's assume that ( f(x) ) is a linear function of the form:

[ f(x) ax b ]

Substituting this into the functional equation, we get:

[ f(f(x)) f(ax b) a(ax b) b a^2x ab b ]

We are given that:

[ a^2x ab b 3x 2 ]

By comparing the coefficients of ( x ) and the constant terms, we can derive the following system of equations:

[ a^2 3 ]

[ ab b 2 ]

These equations will help us determine the values of ( a ) and ( b ).

Resolving the System of Equations

Solving for ( a )

From the equation ( a^2 3 ), we have:

[ a sqrt{3} quad text{or} quad a -sqrt{3} ]

Solving for ( b )

Substituting the values of ( a ) into the second equation ( ab b 2 ):

[ text{Case 1:} quad a sqrt{3} ]

[ (sqrt{3})b b 2 ] [ b(sqrt{3} 1) 2 ] [ b frac{2}{sqrt{3} 1} ] [ b frac{2(sqrt{3} - 1)}{(sqrt{3} 1)(sqrt{3} - 1)} ] [ b frac{2(sqrt{3} - 1)}{3 - 1} ] [ b sqrt{3} - 1 ]

[ text{Case 2:} quad a -sqrt{3} ]

[ (-sqrt{3})b b 2 ] [ b(-sqrt{3} 1) 2 ] [ b frac{2}{1 - sqrt{3}} ] [ b frac{2(1 sqrt{3})}{(1 - sqrt{3})(1 sqrt{3})} ] [ b frac{2(1 sqrt{3})}{1 - 3} ] [ b -sqrt{3} - 1 ]

Final Form of ( f(x) )

Thus, the function ( f(x) ) can take two forms:

[ f(x) sqrt{3}x - (sqrt{3} - 1) ]

[ f(x) -sqrt{3}x - (sqrt{3} 1) ]

This concludes the solution to the functional equation.

Understanding the process and verifying each step can be crucial in solving functional equations. The key steps involve correctly substituting the assumed form of the function, solving the resulting system of equations, and ultimately finding the desired function.