Problem 42 - 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. $$ y=|x| $$

Step-by-Step Solution

Answer

Yes, \(y = |x|\) defines y to be a function of x, as each x maps to exactly one y.

Step 1: Understand a Function

To determine if an equation defines a function, recall that a function assigns exactly one output (y) to each input (x). For each x-value, there must be one and only one corresponding y-value.

Step 2: Analyze the Equation

The given equation is \(y = |x|\). The absolute value function assigns the positive magnitude of x to y, regardless of the sign of x. This means it takes each input x and returns a single output y.

Step 3: Check Specific Values

Test specific values of x: - If \(x = 3\), then \(y = |3| = 3\).- If \(x = -3\), then \(y = |-3| = 3\).For both inputs, there is a unique output, satisfying the definition of a function.

Step 4: Apply the Vertical Line Test

Sketch the graph of \(y = |x|\). If a vertical line crosses the graph at more than one point for any x-value, the relation is not a function. However, each vertical line for any x-value only meets the graph once, confirming it's a function.

Step 5: Conclude

Since each x-value maps to only one y-value and the graph of \(y = |x|\) passes the vertical line test, we conclude that the equation defines y to be a function of x.

Key Concepts

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line. In mathematical terms, the absolute value is always a non-negative number. For any real number \( x \), the absolute value is represented as \( |x| \). This notation means that:

• If \( x \) is positive or zero, then \( |x| = x \).
• If \( x \) is negative, then \( |x| = -x \).

This property is what makes the absolute value function unique. It ensures that no matter whether \( x \) is positive or negative, the output (\( y \) in the equation \( y = |x| \)) remains positive or zero. This characteristic simplifies many algebraic expressions and makes the absolute value a fundamental concept in mathematics. As the function \( y = |x| \) assigns exactly one \( y \)-value for each \( x \)-value, it congruently meets the definition of a mathematical function.

Using the Vertical Line Test

The vertical line test is a straightforward visual method for verifying whether a curve on the graph represents a function. The test states that a curve is a graph of a function if no vertical line intersects the curve more than once at any given \( x \)-coordinate.When you apply this test to the equation \( y = |x| \), you'll see that for every vertical line drawn over any \( x \)-value, the line crosses the graph only once. This means every possible \( x \) corresponds to a single \( y \). It verifies that \( y = |x| \) is a function.
This technique is particularly useful because it provides a quick check: if any vertical line crosses your curve multiple times, the graph does not describe a function. But for \( y = |x| \), the V-shaped graph will always meet our evaluations, showcasing its function status succinctly.

Defining a Function

A \( function \) is a special kind of relation where each input value has exactly one output value.

• The key rule for functions is that for each \( x \)-value (input), there must be exactly one \( y \)-value (output).
• This assures no ambiguity: every time you input a certain \( x \), you know exactly what \( y \) will be.

In the context of the given problem, the equation \( y = |x| \) perfectly satisfies the criteria of a function because:

• It consistently provides a single output \( y \) for each possible \( x \).
• Whether \( x \) is positive, negative, or zero, \( |x| \) computes to one definite, predictable \( y \).

This fittingly aligns with the function definition, ensuring clarity and precision in mathematical operations involving \( y = |x| \). Functions are foundational in algebra and calculus, helping to map relationships and predict outcomes accurately.