Problem 44 - 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. $$ x+1=|y| $$

Step-by-Step Solution

Answer

The equation does not define \(y\) as a function of \(x\). For \(x = 0\), the pairs \((0, 1)\) and \((0, -1)\) show why.

Step 1: Understanding the Problem

We need to determine if the equation \(x+1=|y|\) defines \(y\) as a function of \(x\). In a function, each value in the domain (\(x\)) should correspond to exactly one value in the range (\(y\)).

Step 2: Analyzing the Absolute Value Equation

The equation \(x + 1 = |y|\) involves an absolute value. Recall the definition of absolute value: if \(|y| = c\), then \(y = c\) or \(y = -c\). Thus, if \(x + 1 = a\), \(y\) could be \(y = a\) or \(y = -a\).

Step 3: Solving for Multiple Values of y

Set \(x + 1 = |y|\) such that \(|y| = a\), we have:\[ y = x + 1 \quad \text{and} \quad y = -(x + 1) \] This means for every \(x\), \(y\) could potentially be two values: \(x + 1\) and \(-x - 1\).

Step 4: Conclusion on Function Definition

Since each value of \(x\) gives us two possible values for \(y\), \(y\) is not a single-valued output for each input \(x\). Therefore, \(x+1=|y|\) does not define \(y\) as a function of \(x\).

Step 5: Finding Two Ordered Pairs

Choose a specific value for \(x\). Let \(x = 0\):- If \(x = 0\), then \(x + 1 = 1\), so \(|y| = 1\).- So, \(y = 1\) or \(y = -1\).- This gives ordered pairs \((0, 1)\) and \((0, -1)\), showing more than one value of \(y\) for a single value of \(x\).

Key Concepts

Absolute Value Equations

An absolute value equation involves expressions within absolute value brackets, such as \(|y|\). The absolute value of a number is its distance from zero on the number line, which is always non-negative.
For example, \(|-3| = 3\) and \(|3| = 3\). So, absolute value ignores whether a number is negative or positive; it simply focuses on magnitude.

When solving an equation like \(|y| = c\), we know two things may be true:

• \(y = c\)
• \(y = -c\)

This is because both \(+c\) and \(-c\) are at the same distance from zero. In an equation like \(x + 1 = |y|\), \(y\) can take on two possible values for any given \((x + 1)\): one positive and one negative. Therefore, absolute value equations often present two solutions.

Domain and Range

In algebra, the **domain** refers to all possible input values (commonly represented as \(x\)) that a function can accept. The **range** refers to the resulting set of possible output values (usually denoted by \(y\)).
To qualify as a function, each input from the domain must map to exactly one output in the range.

Consider the equation \(x + 1 = |y|\). If each \(x\) corresponds to two possible \(y\) values due to the nature of absolute values (i.e., \((y = x + 1)\) or \((y = -(x + 1))\)), then the relation does not define \(y\) as a function of \(x\).

For example, if \(x = 0\), we obtain \(y = 1\) and \(y = -1\), indicating that \(x\) is not paired with a unique \(y\) value, thereby violating the one-to-one requirement of a function.

Ordered Pairs

In algebra, an **ordered pair** is a fundamental way to depict relationships between two coordinated elements, usually noted as \(x\) and \(y\). Ordered pairs are expressed in the form \((x, y)\).
Each pair consists of an input \(x\) (first element) and an output \(y\) (second element).

For instance, in the equation \(x + 1 = |y|\), if \(x = 0\), we derive two possible values for \(y\): \((0, 1)\) and \((0, -1)\). This indicates that a single value of \(x\) leads to two outputs for \(y\), which mathematically illustrates why \(y\) is not a function of \(x\) in this context.

Ordered pairs help visually illustrate whether a relation represents a function—only if each \(x\) value maps to a single \(y\) value can the relation be considered a function. When one \(x\) has multiple \(y\) outputs, this uniqueness requirement is breached.