Solving for f1 in Complex Function Equations

Mathematical puzzles and equations often present intricate challenges, requiring a keen mind and a solid understanding of algebraic principles. In this article, we delve into a specific problem involving function equations and demonstrate a step-by-step method to solve for f1. These types of equations are not only intellectually stimulating but also serve as excellent practice for those looking to enhance their mathematical problem-solving skills.

Introduction to the Problem

The problem at hand involves determining the value of f1 given the equation f(x)3f(1-x)x2. This equation requires us to find the value of f1 through a series of logical steps that involve substitution and algebraic manipulation.

Solving for f1 with Substitution

To solve for f1, we can use substitution. Given the equation f(1)3f(0), we substitute x 0 and x 1 to create a system of equations:

Substituting x 1

Using the equation f13f0:

When x 1, the equation becomes f13f0-10, which simplifies to f0-f1 13.

Substituting x 0

Using the equation f03f10:

When x 0, the equation becomes f03f10, which simplifies to f0-3f1.

Combining the Equations

Equating the two expressions for f0 from steps 1 and 2, we get:

f1 13-3f1.

Solving for f1 from the equation f1 1-9f1:

f1-9f1-1.

Therefore, -8f1-1.

f118.

General Solution for fA

The method to solve for fA can be generalized as follows:

Solving for fA in the General Form

Given the equation fA3f1-A-A2, we can set x A and x 1-A to derive a general solution for fA and f1-A.

When x A, we have:

fA3f1-A-A2

When x 1-A, we have:

f1-A3fA-1-A2

Solving for fA and f1-A, we get:

fA-14A2 34A-38

Special Case for f1

For the specific case when A 1, we substitute and simplify to find:

f1-14 34-38

f118

Conclusion

Solving for f1 in such function equations requires a systematic approach and a good understanding of algebraic manipulation. The process outlined above not only provides a clear solution but also serves as a valuable tool for tackling similar problems in the future. This type of mathematical puzzle not only enhances problem-solving skills but also fosters a deeper appreciation for the elegance and complexity of mathematics.