Problem 38 - 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. $$ y^{2}=x $$

Step-by-Step Solution

Answer

The equation does not define \( y \) as a function of \( x \); example pairs: (4, 2) and (4, -2).

Step 1: Understand the Problem

We need to determine if the equation \( y^2 = x \) defines \( y \) as a function of \( x \). A function has the property that each input value maps to exactly one output value.

Step 2: Analyze Function Definition

For \( y \) to be a function of \( x \), any given \( x \) must yield a unique \( y \). Thus, for a fixed \( x \), there should not be more than one corresponding \( y \) value.

Step 3: Substitute Values

Try substituting a specific \( x \) value into the equation to see if multiple \( y \) values fulfill the equation. For example, let \( x = 4 \).

Step 4: Solve for y Values

Substitute \( x = 4 \) into the equation: \( y^2 = 4 \). Take the square root of both sides to find \( y \): \( y = 2 \) or \( y = -2 \).

Step 5: Conclusion on Function Definition

Since substituting \( x = 4 \) gives two possible \( y \) values, 2 and -2, the equation does not define \( y \) as a function of \( x \).

Step 6: Identify Ordered Pairs

The ordered pairs \((4, 2)\) and \((4, -2)\) demonstrate that one \( x \)-value corresponds to multiple \( y \)-values.

Key Concepts

Function Definition

In mathematics, a function is a fundamental concept that refers to the relationship between two sets, where each element from the first set (called the domain) is associated with exactly one element from the second set (called the codomain). To determine if an equation defines a function, we check whether each input value (often represented as the variable \(x\)) is mapped to only one output value (represented by \(y\)).

For instance, in the equation \(y^2 = x\), we want to verify if any given \(x\) results in a unique \(y\). If for even one \(x\) there is more than one corresponding \(y\), then \(y\) is not a function of \(x\).

The principle here is often called the "vertical line test" where, if drawing vertical lines through the graph of the equation shows any line intersecting the graph at more than one point, then the equation does not define a function.

Ordered Pairs

Ordered pairs are a fundamental concept when discussing relations and functions in algebra. An ordered pair is expressed as \((x, y)\), where \(x\) represents the first value, typically the input or independent variable, and \(y\) represents the second value, typically the output or dependent variable.

When evaluating function definitions, it's crucial to see if one \(x\)-value can lead to multiple \(y\)-values. Consider the equation \(y^2 = x\) analyzed earlier, which resulted in a situation where input \(x = 4\) could map to two outputs, creating two ordered pairs: \((4, 2)\) and \((4, -2)\).

These two pairs exemplify that the input \(4\) does not correspond to a single \(y\), implying that the equation cannot be considered a function.

Analyzing Equations

Analyzing equations in the context of functions involves dissecting the relationship between variables to determine whether each input yields a single output. This often involves substituting specific values into the equation and solving for the other variable to test the function rule.

For example, substituting \(x = 4\) in the equation \(y^2 = x\), we arrive at \(y^2 = 4\), which upon solving yields \(y = 2\) and \(y = -2\).

This step reveals the equation's nature as not defining \(y\) as a function of \(x\) since an input of \(4\) results in multiple outputs.

Such analysis can include using graphical methods, like plotting the graph of the equation, or algebraic techniques, like substituting values, to comprehensively understand the equation and its implications for functions.