Understanding the Function and its Derivatives

In mathematics, the study of functions, particularly polynomial and rational functions, is fundamental. This article will guide you through a complex problem involving a specific function. We will use algebraic manipulation to determine the value of f(3) when given the function in the form: f(x) x^6 frac{1}{x^6}. We'll begin by breaking down the solution into manageable steps and deepening our understanding of how to handle such problems.

Step-by-Step Solution

The initial step in working with the function f(x) x^6 frac{1}{x^6} is to derive key equations from the given information. Given that x frac{1}{x} 3, we will use this information to complete the necessary algebraic manipulations. Our goal is to find the specific value of f(3).

Step 1: Squaring the Given Equation

First, we need to square the given equation x frac{1}{x} 3.

(1) Square both sides of the equation:

[(x frac{1}{x})^2 3^2]

(2) Expand the left side using the formula (a b)^2 a^2 2ab b^2:

[x^2 2 frac{1}{x^2} 9]

(3) Isolate the term involving (frac{1}{x^2}):

[x^2 frac{1}{x^2} 7]

Step 2: Finding (x^2 frac{1}{x^2}) and (x^3 frac{1}{x^3})

Next, we calculate (x^3 frac{1}{x^3}) using the values obtained in the previous step.

(1) Multiply both the initial equation x frac{1}{x} 3 with ((x frac{1}{x})):

(2) We know that (x^2 frac{1}{x^2} 7), so:

(3) Combine the results to find (x^3 frac{1}{x^3}):

[(x frac{1}{x})(x^2 frac{1}{x^2} - 1) x^3 frac{1}{x^3} - (x frac{1}{x})]

(4) Substitute the known values:

[3 cdot (7 - 1) - 3 x^3 frac{1}{x^3} - 3]

(5) Simplify the expression:

[3 cdot 6 - 3 18]

Step 3: Strengthening the Function

Using the calculated values from the previous steps, we can now derive the value for f(3) in the function (f(x) x^6 frac{1}{x^6}).

(1) Square the value of (x^3 frac{1}{x^3} 18):

[(x^3 frac{1}{x^3})^2 18^2]

(2) Expand using the formula (a b)^2 a^2 2ab b^2:

[x^6 2 frac{1}{x^6} 324]

(3) Isolate the term involving (x^6 frac{1}{x^6}):

[x^6 frac{1}{x^6} 324 - 2]

(4) Simplify the expression:

[x^6 frac{1}{x^6} 322]

Thus, the value of f(3) is 322.

Conclusion

The journey to solving for f(3) with the function f(x) x^6 frac{1}{x^6}, given x frac{1}{x} 3, involved careful algebraic manipulation and step-by-step derivations. By first finding the values of x frac{1}{x} 3, (x^2 frac{1}{x^2} 7), and (x^3 frac{1}{x^3} 18), we successfully determined the value of x^6 frac{1}{x^6} 322. This exercise not only demonstrates the power of algebraic techniques in solving complex problems but also highlights the importance of polynomial functions and their properties.

In conclusion, when tackling such algebraic problems, breaking down the equation into coherent steps and using fundamental algebraic principles can lead you to the correct solution. Understanding and mastering these techniques will undoubtedly enhance your problem-solving skills in both mathematics and beyond.

Keywords

function simplification, algebraic manipulation, polynomial functions