Problem 16 - Contemporary Precalculus

Question

Determine whether the equation defines \(y\) as a function of \(x\) or defines \(x\) as a function of \(y\) $$y-4 x^{3}-14=0$$

Step-by-Step Solution

Answer

Answer: The given equation defines \(y\) as a function of \(x\). The function can be represented as \(y = f(x)\) where \(f(x) = 4x^3 + 14\).

Step 1: Solve for y in terms of x

To solve for \(y\) in terms of \(x\), we need to isolate \(y\) on one side of the equation. We have: $$y - 4x^3 - 14 = 0$$ Add \(4x^3\) and \(14\) to both sides of the equation to isolate \(y\): $$y = 4x^3 + 14$$ Since we can uniquely express \(y\) in terms of \(x\), it means the given equation defines \(y\) as a function of \(x\). This function can be represented as \(y = f(x)\) where \(f(x) = 4x^3 + 14\).

Step 2: Attempt to solve for x in terms of y

To solve for \(x\) in terms of \(y\), we need to isolate \(x\) on one side of the equation. However, given the equation \(y = 4x^3 + 14\), it is difficult to express \(x\) as a function of \(y\) without using any special functions (such as the cube root function). Moreover, the primary aim of the exercise is to determine whether the equation defines \(y\) as a function of \(x\) or defines \(x\) as a function of \(y\), which has already been determined in Step 1. In conclusion, the given equation \(y - 4x^3 - 14 = 0\) defines \(y\) as a function of \(x\): \(y = f(x) = 4x^3 + 14\).

Key Concepts

Expressing Variables

Expressing variables is the process of reorganizing an equation so that one specific variable is isolated on one side. This helps in understanding the relationship between different variables in an equation.

Let's consider our equation:

• \(y - 4x^3 - 14 = 0\)

In this exercise, we need to express \(y\) in terms of \(x\). To do this, we isolate \(y\) by adding \(4x^3\) and \(14\) to both sides:

• \(y = 4x^3 + 14\)

This means that \(y\) is now easily expressed as a function of \(x\). This technique, expressing one variable in terms of another, simplifies understanding and allows us to analyze the behavior of the function more directly. It is fundamental in mathematics for solving equations and graphing functions.

Solving Equations

Solving equations is all about finding the value of variables that satisfy a given equation. In the equation \(y - 4x^3 - 14 = 0\), we aimed to determine whether the equation could uniquely define \(y\) as a function of \(x\), or vice versa.

By solving for \(y\), we successfully find:

• \(y = 4x^3 + 14\)

But what happens when we try to solve for \(x\) in terms of \(y\)? To do this, we begin with:

• \(y = 4x^3 + 14\)

Solving for \(x\) involves isolating it, which means rearranging the equation to:

• \(4x^3 = y - 14\)
• \(x^3 = \frac{y - 14}{4}\)

And finally:

• \(x = \sqrt[3]{\frac{y - 14}{4}}\)

However, solving for \(x\) in this way introduces complexities such as cube roots, demonstrating that while technically possible, expressing \(x\) as a function of \(y\) is less straightforward than the initial intent of expressing \(y\) as a function of \(x\). This highlights the importance of choosing the right approach based on the given equation and what is required from it.

Function Notation

After expressing \(y\) in terms of \(x\), we are able to use function notation to represent the equation in a more streamlined form. Function notation involves expressing equations in a way that clearly indicates the output of a function based on a given input.

For instance, once we have determined that \(y = 4x^3 + 14\), we can express this using function notation as:

• \(y = f(x)\)

where

• \(f(x) = 4x^3 + 14\)

This notation clearly shows that \(y\), the output, is dependent on \(x\), the input through the function \(f\). It simplifies analyzing and graphing the relationship between the variables, underscoring the advantage of expressing equations in function form.

Function notation is particularly useful because it:

• Makes equations easier to read and manage.
• Provides a clear framework for understanding how changes in one variable affect the other.

By using function notation, you enhance the clarity and functionality of your mathematical expressions.