Solving and Simplifying the Function x - 3/x x - 1/2x

Introduction

In this article, we delve into the process of solving and simplifying a complex algebraic function, namely x - 3/x x - 1/2x. This type of function is both a common challenge in algebra and a fundamental concept in understanding more advanced mathematical ideas. We will explore the steps to solve for x in terms of given variables and simplify the resulting expressions. By the end, readers will have a clear understanding of how to approach similar problems and appreciate the elegance of algebraic manipulation.

Step-by-Step Solution

Method 1: Using Substitution with u x - 3/x

Let's start with the substitution method, where we let u x - 3/x. Then, we have:

u x - 3 xu - 1 -u^3 x ((u^3) 1)/(u)

Substituting u x - 3/x into the above equation, we get:

x ((x - 3/x)^3 1)/(x - 3/x)

Further simplification involves rearranging and substituting values, which can be quite complex. However, the primary goal is to reach a simplified form.

Method 2: Direct Substitution with y x - 3/x

Another approach is to let y x - 3/x. This gives us:

xy x - 3 x - yx 3/y x(1 - y) 3/y x 3/y / (1 - y)

Substituting y x - 3/x into this equation, we get:

x 3/(x - 3/x) / (1 - (x - 3/x))

Further simplification leads to:

x x^3 / (x^3 - 3x 3)

Method 3: Using the Variable Substitution t (x - 3)/x

Let's now consider the substitution t (x - 3)/x. This gives us:

t (x - 3)/x x (t^3)/(1 - t)

Substituting x (t^3)/(1 - t) into the function, we have:

f(t) 1/2(1 - 1/x) 1/2(1 - (1-t)/(t^3)) f(t) t^3/(2t^3) f(t) 1/2

Converting back to x, we find that:

f(x) x^3 / (2x^3)

Which simplifies to:

f(x) 1/x

Conclusion and Further Exploration

In conclusion, by using various substitution methods, we have successfully simplified the given function x - 3/x x - 1/2x to 1/x. Each step involves careful algebraic manipulation, highlighting the importance of persistent problem-solving in mathematics.

Keywords

function simplification, algebraic manipulation, exponential expressions