Problem 39 - Elementary and Intermediate Algebra

Question

Determine whether each equation defines \(y\) to be a function of \(x .\) If it does not, find two ordered pairs where more than one value of \(y\) corresponds to a single value of \(x .\) See Example 3. $$ y^{4}=x $$

Step-by-Step Solution

Answer

The equation does not define \(y\) as a function of \(x\) since one \(x\) can give two \(y\) values, like (1, 1) and (1, -1).

Step 1: Understand the definition of a function

A relation defines a function when each input (\(x\) value) has exactly one output (\(y\) value). For every \(x\), there should be at most one \(y\).

Step 2: Analyze the given equation

The equation given is \(y^4 = x\). We need to check if one \(x\) always corresponds to exactly one \(y\). If it doesn't, then \(y\) is not a function of \(x\).

Step 3: Solve the equation for \(y\)

In the equation \(y^4 = x\), solving for \(y\) gives \(y = \pm \sqrt[4]{x}\). Thus, there are potentially two \(y\) values (positive and negative fourth roots) for a single \(x\).

Step 4: Identify if the relation is a function

Since for any positive \(x\), there are two possible \(y\) values (\(y = \sqrt[4]{x}\) and \(y = -\sqrt[4]{x}\)), the given equation does not define \(y\) as a function of \(x\).

Step 5: Find two ordered pairs

If \(x = 1\), then \(y = \pm 1\), resulting in ordered pairs (1, 1) and (1, -1). These show that a single \(x\) value corresponds to two different \(y\) values.

Key Concepts

Understanding Relation and Function

A key concept in algebra is understanding the difference between a 'relation' and a 'function'. Both involve pairs of numbers, but their definitions hold a slight difference which is pivotal in solving equations. A **relation** is simply a set of ordered pairs. This means it can associate every first element (usually denoted as \(x\)) with one or more second elements (usually denoted as \(y\)). In contrast, a **function** is a special kind of relation. It requires that each \(x\) links to exactly one \(y\).
This distinction is crucial when analyzing mathematical equations. If you have an equation and you want to determine if it represents a function, you must ensure every input has only one output. If any \(x\) value corresponds to two or more \(y\) values, it's a relation, not a function. This is often visualized with the "vertical line test" on a graph, where a vertical line must intersect the graph at most once to qualify as a function.

Exploring Ordered Pairs

An ordered pair is a pair of elements written in a specific order, typically as \((x, y)\).
Ordered pairs are foundational in understanding relations and functions because they express the precise connection between the two values. In the coordinate system, the first number, \(x\), often represents the horizontal position, while \(y\) represents the vertical position.
Understanding ordered pairs is essential in determining whether an equation is a function. For instance, in the given equation \(y^4 = x\), solving for different \(x\) values can result in multiple \(y\) values. If \(x = 1\), \(y\) could be 1 or -1. This creates two different ordered pairs: (1, 1) and (1, -1).
Such disjointed pairs violate the definition of a function, as one input results in multiple outputs. By examining ordered pairs, you can easily decide if a relation qualifies as a function.

Steps Involved in Solving Equations

Solving equations is a critical skill in algebra. It involves manipulating a given equation to find the unknown variable, typically denoted by \(y\) or \(x\). To solve an equation effectively, follow these steps:

• First, **understand the problem**. Grasp the mathematical concept behind the relation. Is it solving for \(y\) or \(x\)?
• Next, **rearrange the equation** to isolate the variable you are solving for. This often involves algebraic manipulations such as addition, subtraction, multiplication, or division.
• In some cases, you might need to apply more advanced operations, such as factoring or dealing with roots, like in the equation \(y^4 = x\), where solving for \(y\) gives \(y = \pm \sqrt[4]{x}\).
• Finally, **verify your solution** by substituting values back into the original equation to ensure accuracy.

These steps not only provide a solution, but they also reassure that the solution fits within the context of the equation being a function or merely a relation, based on the output possibilities for each input.