Exploring the Curve of 3fx-1 with fx x^3

This article delves into the intricate process of transforming a given equation, specifically focusing on the algebraic manipulation of functions to uncover the equation of a new curve. We will explore the process of changing fx x^3 into the equation of the curve 3fx-1, demonstrating a step-by-step approach and the underlying mathematical principles.

Introduction to the Problem

The given problem revolves around the transformation of the function fx x^3 into a new form, specifically the equation of the curve 3fx-1. This type of problem is commonplace in algebra and pre-calculus mathematics, and understanding the steps can be crucial for students and professionals alike.

Understanding the Function fx x^3

Let us begin by assessing the given function fx x^3. This third-degree polynomial function has the following properties:

Domain and Range

Domain: All real numbers ({eq}mathbb{R}{/eq}) as there are no restrictions on the input. Range: All real numbers ({eq}mathbb{R}{/eq}) since the function covers the entire set of real numbers.

Graphical Representation

The graph of fx x^3 is a continuous curve with a distinctive 'S' shape, passing through the origin (0,0). The curve is symmetric with respect to the origin, indicating that it is an odd function.

Manipulating the Given Equation to 3ffx-1

Step-by-Step Process

The key to understanding the transformation is to identify how the operations affect the equation. Let's break down the steps:

Step 1: Simplify fx-1

Starting with the given equation, fx-1 x-1x-1^3. This can be simplified further as follows:

fx-1 x-1^3 x-1(x^2 3x 2) x^3 - 3x^2 3x - 1

This step involves expanding and simplifying the given function fx x^3 into the form fx-1. This simplification is crucial for identifying the behavior of the transformed function.

Step 2: Multiply by 3

Next, we need to transform fx-1 into 3fx-1. This involves multiplying the entire simplified form by 3:

3fx-1 3(x^3 - 3x^2 3x - 1) 3x^3 - 9x^2 9x - 3

This step showcases how scaling a function changes its nature, including its amplitude and curvature.

Deriving the Equation of the Curve 3fx-1

Final Equation

From the above step-by-step transformation, we arrive at the final equation of the curve:

3fx-1 3x^3 - 9x^2 9x - 3

This equation is the transformed version of the original function, showcasing the algebraic intricacies at play in such transformations.

Conclusion and Further Insights

Understanding how to transform functions, particularly polynomial functions, is a fundamental aspect of algebra and pre-calculus. The process of transforming fx x^3 into 3fx-1 not only highlights the importance of algebraic manipulation but also provides insight into the graphical representation of the resulting curve.

For students and professionals, grasping these concepts is essential for tackling more complex problems in mathematics and related fields. The transformation of functions is not only a theoretical exercise but also has practical applications in fields such as engineering, physics, and economics.