Problem 68 - Algebra and Trigonometry

Question

Determine whether the equation defines y as a function of x. (See Example 9.) \(x=y^{4}\)

Step-by-Step Solution

Answer

The equation does not define \(y\) as a function of \(x\) because \(y\) is not uniquely determined.

Step 1: Understanding the Problem

To determine if an equation defines a function, we need to check if for each input value of one variable, there is exactly one output value of the other variable. Here, the equation is \(x = y^4\), and we need to see if \(y\) is uniquely determined for each \(x\).

Step 2: Solving for y in terms of x

Since the equation is \(x = y^4\), we solve for \(y\) to express it in terms of \(x\). This gives \(y = \pm \sqrt[4]{x}\), meaning \(y\) could be either the positive fourth root or the negative fourth root of \(x\).

Step 3: Analyzing the Uniqueness of y

For a given \(x\), there are two possible values of \(y\), \(y = \sqrt[4]{x}\) and \(y = -\sqrt[4]{x}\). This means that for a given value of \(x\), \(y\) is not uniquely determined, as it has two possible values.

Step 4: Conclusion

Since \(y\) is not uniquely determined for each value of \(x\), the equation \(x = y^4\) does not define \(y\) as a function of \(x\).

Key Concepts

Solving Equations

When faced with the task of solving equations, the primary aim is to find the value of one variable in terms of another. Here, we start with the equation \(x = y^4\). Our goal is to express \(y\) in terms of \(x\). By rearranging the equation, we arrive at \(y = \pm \sqrt[4]{x}\).
This solution gives us the value of \(y\) as both the positive and negative fourth roots of \(x\).
Solving equations in this way requires us to consider all possible solutions to ensure no value is overlooked.

It is important to apply correct algebraic transformations, such as taking roots or logarithms, to isolate the variable of interest. Such transformations must be done carefully to retain all potential solutions, especially when dealing with even roots, which can yield more than one possible outcome.

Uniqueness Analysis

A function requires each input to correspond to exactly one output. In uniqueness analysis, it is crucial to check if every input value results in one and only one output value. For our equation \(x = y^4\), we observed that for a given \(x\), there are two possible values for \(y\): \(y = \sqrt[4]{x}\) and \(y = -\sqrt[4]{x}\).
This means, if \(x\) is positive, \(y\) can be both positive and negative. Hence, \(y\) fails the test for uniqueness because it does not map back to a single value for each \(x\).

Uniqueness is a critical property in defining functions. It helps in understanding whether a relationship between variables fits the mathematical definition of a function. Without uniqueness, the relationship cannot be considered a function.

Expressing Variables in Terms of Others

Expressing one variable in terms of another involves solving an equation to isolate a variable. In the case of our exercise, we express \(y\) in terms of \(x\) from \(x = y^4\) as \(y = \pm \sqrt[4]{x}\).
This approach highlights the interdependent nature of variables, where one can be described fully or partly using another variable or a combination of variables.

Expressing variables in terms of others is a fundamental technique in algebra, allowing us to better understand the relationships and dependencies between different variables within mathematical models. This technique is essential for solving more complex equations and is foundational in calculus and beyond.