Exploring the Polynomial Function f(x) Given Conditions

In this article, we delve into the problem of finding a polynomial function f(x) such that when composed with itself twice, the result is a function of the form 3x2. Specifically, we will determine the values of a and b in the general polynomial fx axb that satisfy this condition.

Rational Behind the Solution

We start by assuming that fx is a polynomial function in the form:

fx axb, where a and b are constants.

When we compose fx with itself, we obtain:

ffx a(axb)b a2xab

We are given that:

ffx 3x2

From this, we can derive the following equations by comparing the coefficients:

1. a2 3 ? a ±√3

2. ab 2

Deriving the Values of a and b

From the first equation, we have two possible values for a:

a √3 or a -√3

Substituting these values into the second equation, we get:

Case 1: a √3

ab 2 ? b 2 / (1 √3) (2(1 - √3)) / (1 - 3) (2 - 2√3) / -2 1 - √3

Case 2: a -√3

ab 2 ? b 2 / (1 - √3) (2(1 √3)) / -2 -1 - √3

Thus, the polynomial functions that satisfy the given condition are:

fx √3x^(1 - √3)

or

fx -√3x^(-1 - √3)

Verification

To verify, let's compute the composition of each function with itself:

1. For fx √3x^(1 - √3):

ffx (√3x^(1 - √3))(√3x^(1 - √3)) 3x^(2 - 2√3)

Since 2 - 2√3 ≠ 2, this function does not satisfy the given condition.

2. For fx -√3x^(-1 - √3):

ffx (-√3x^(-1 - √3))(-√3x^(-1 - √3)) 3x^(2 2√3)

Since 2 2√3 ≠ 2, this function also does not satisfy the given condition.

The correct approach is to consider the simpler form of the function, where the polynomial is affine (i.e., linear):

Case 1: fx √3x - √3

ffx (√3x - √3)(√3x - √3) 3x2 - 2√3x 3

Case 2: fx -√3x √3

ffx (-√3x √3)(-√3x √3) 3x2 - 2√3x 3

Both cases yield the correct result, confirming that the function fx -√3x √3 is a valid solution.

Conclusion

The polynomial function fx -√3x √3 satisfies the given condition. This solution demonstrates the process of solving polynomial function compositions and highlights the importance of careful algebraic manipulation in finding the correct solution.