Understanding the Function f(x) (x/(1-x))^2 and Its Applications

This article delves into the function f(x) (x/(1-x))^2, exploring its derivation from other forms and its applications in algebraic manipulation. We will provide clear explanations and examples to ensure a thorough understanding of this mathematical concept.

Derivation of the Function

1. Introducing the Function

The function in question is given by:

f(x) (x/(1-x))^2

2. From f(x/x1) x^2 to f(x) (x/(1-x))^2

Let’s consider the transformation from f(x/x1) x^2.

To begin, let's set:

u x/x1

This implies:

x u*x1

Substituting u back in the equation:

u * u * x1 u^2 * x1

And:

u x - u*x1

Thus:

x u / (1 - u)

Using this, the original function can be rewritten as:

f(u) (u / (1 - u))^2

Replacing u back with x, we get:

f(x) (x / (1 - x))^2

3. Verification of the Transformation

To verify, we can check the original transformation:

f(x/x1) (x/x1) / (1 - x/x1)

Simplifying the right-hand side:

(x/x1) / ((x1 - x) / x1)

(x/x1) * (x1 / (x1 - x))

(x / (x1 - x))^2

This confirms that our transformation is correct.

Another way to represent the function is using a variable substitution:

Let:

u x / x1

Then:

x u * x1

The function becomes:

fu (u / (1 - u))^2

Substituting u back with x, we get:

fx (x / (1 - x))^2

4. Simplified Transformation Approach

A more instructive method is to directly substitute:

u x / (x - 1)

This implies:

fu (u / (1 - u))^2

Substituting u back with x, the function becomes:

fx (x / (1 - x))^2

5. Understanding the Function Mechanism

The function can be broken down as follows:

Alternatively, let’s consider:

u x / (x - 1)

Then:

fu u^2 / (u - 1)^2

Simplifying, we get:

fu (u / (1 - u))^2

6. Verification and Application

To verify, let's use the substitution:

(x1 / (x - 1))^2

This means:

u * (1 / (1 - u))^2

Simplifying, we get:

fu (u / (1 - u))^2

Replacing u with x, the function becomes:

fx (x / (1 - x))^2

This confirms that our derived function is correct and useful for further algebraic manipulations and applications in mathematics.

Conclusion

This article has provided a comprehensive understanding of the function f(x) (x / (1 - x))^2, detailing its derivation and application. Whether through direct substitution or simplified transformations, the function's utility and clarity are evident, making it a valuable tool in various mathematical operations.