Problem 37 - 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 \(x .\) $$ x+1=|y| $$

Step-by-Step Solution

Answer

The equation does not define \(y\) as a function of \(x\).

Step 1: Analyze the Equation

We begin by evaluating the given equation \(x + 1 = |y|\). The absolute value symbol \(|y|\) indicates that regardless of whether \(y\) is positive or negative, \(x+1\) will be the same as it outputs a non-negative value for \(|y|\).

Step 2: Solve for \(y\)

To solve for \(y\), consider that \(|y| = a\) implies that \(y\) could equal \(a\) or \(-a\). Thus, \(y = x + 1\) or \(y = -(x + 1)\). This shows two possible solutions for \(y\) when given a single \(x\).

Step 3: Find Ordered Pairs

Choose a specific value for \(x\) to demonstrate how different \(y\) values correspond to the same \(x\). For example, if \(x = 2\), then \(x + 1 = 3\). Consequently, \(y = 3\) or \(y = -3\). Thus, the points are \((2, 3)\) and \((2, -3)\).

Step 4: Conclusion of Functionality

Since there are two distinct \(y\) values for a single \(x\) value (i.e., \(x = 2\)), the equation does not define \(y\) to be a function of \(x\). For a relation to be a function, each \(x\) must have exactly one corresponding \(y\) value.

Key Concepts

Understanding Ordered Pairs in Functions

Ordered pairs are fundamental building blocks in mathematics. They consist of two elements placed in a specific order, usually represented as \((x, y)\). The first element is typically the x-coordinate, and the second is the y-coordinate. These pairs are used to denote points in a coordinate plane, tracing the path of a graph.

• The x-value is called the "input" or "independent variable."
• The y-value is referred to as the "output" or "dependent variable."

In the context of functions, each x-value should correspond to only one y-value, to ensure that a relation is indeed a function. When a single x-value links to multiple y-values, it indicates a failure to define a function, as seen in the example with ordered pairs \((2, 3)\) and \((2, -3)\). This creation of different y-values for the same x signifies that the relation is not a function.

Exploring the Absolute Value Concept

The absolute value of a number is a measure of its distance from zero on the number line, always resulting in a non-negative number. In mathematical terms, the absolute value is denoted by two vertical bars around the number, like \(|y|\).

• For any positive number or zero, \(|y| = y\).
• For any negative number, \(|y| = -y\), which converts it to a positive value.

Understanding how absolute value impacts equations such as \(x+1 = |y|\) is crucial. The absolute value allows y to be either \(x + 1\) or \(- (x + 1)\), because both lead to the same absolute result \(x+1\). This dual outcome demonstrates why a single x-value can yield more than one y-result, influencing the determination of function relationships.

Mastering the Art of Solving Equations

Solving equations involves finding the values of variables that make the equation true. It's like unlocking a puzzle where every operation allows you to move closer to the solution.

• Begin by simplifying the equation, gathering like terms, and reducing fractures.
• Apply inverse operations to isolate the variable of interest.
• For absolute value equations like \(x + 1 = |y|\), set up two distinct equations: one equating \(y\) to the positive expression, and another to the negative.

In our example, this process shows why \(y = x + 1\) and \(y = - (x + 1)\). Ensure you check your final solutions by plugging them back into the original equation to verify their validity. Dealing with equations is about practice and understanding how and why different mathematical operations expose the true values of unknowns.