Problem 56 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 55-56: Explain why the graph of the equation is not the graph of a function. $$ x=-|y| $$

Step-by-Step Solution

Answer

The graph fails the vertical line test; thus, it is not a function.

Step 1: Understand the equation

The given equation is \( x = -|y| \). This equation defines \( x \) in terms of the absolute value of \( y \). It means \( x \) is the negative of the absolute value of \( y \).

Step 2: Consider properties of a function

For a relation to be a function, each input \( y \) must have only one output \( x \). In other words, for each \( y \), the equation should produce exactly one value of \( x \).

Step 3: Analyze the absolute value expression

The absolute value \(|y|\) produces a non-negative value for any input \( y \), meaning \(|y|\) is always \( \geq 0 \). Therefore, \( -|y| \) will always be \( \leq 0 \).

Step 4: Investigate the symmetry of the equation

For \( y = a \) and \( y = -a \) (where \( a \geq 0 \)), the expression \(|y|\) yields the same value, implying both \( x = -a \) for \( y = a \) and \( y = -a \). So both positive and negative values of \( y \) map to the same \( x \), creating a vertical alignment.

Step 5: Apply the vertical line test

If a vertical line intersects the graph at more than one point for any value of \( x \), the graph does not represent a function. In this case, because different \( y \) values (viz., \( y = a \) and \( y = -a \)) map to the same \( x \), a vertical line at these \( x \) values intersects the graph at multiple points.

Step 6: Conclude based on observations

Because multiple \( y \) values map to the same \( x \), and vertical lines intersect at multiple points, the graph of \( x = -|y| \) does not satisfy the definition of a function as it does not provide a unique output \( x \) for each input \( y \).

Key Concepts

Vertical Line Test

The vertical line test is a simple method to determine if a graph represents a function. A function can have only one output for each input value. To test this, imagine drawing many vertical lines across the graph.

• If a vertical line hits the graph at more than one spot, then the graph is not a function.
• This is because a vertical line touching the graph in multiple places means there are multiple outputs for a single input, which violates the definition of a function.

In the equation given, \(x = -|y|\), multiple values for \(y\) (such as \(y = a\) and \(y = -a\)) can map to the same \(x\). This results in a vertical line intersecting the graph at both points, proving it's not a function. It's a reliable first check when examining if a graph is a function.

Absolute Value

The absolute value of a number is its distance from zero on the number line and is always a non-negative value. In the equation \(x = -|y|\), the term \(|y|\) indicates that whatever \(y\) is, its absolute value will dictate the value of \(x\).

• Absolute value will always be greater than or equal to zero: \(|y| \geq 0\).
• Thus, \(-|y|\) will always be less than or equal to zero: \(-|y| \leq 0\).

This characteristic of absolute value leads to significant implications in the context of functions. Since \(|y|\) is unaffected by the sign of \(y\), both a positive and a negative \(y\) will yield the same absolute value, resulting in overlapping points on the graph. Thus, a single \(x\) value in the equation \(x = -|y|\) can have multiple corresponding \(y\) values.

Graph Symmetry

Symmetry in graphs can help determine if a curve matches the definition of a function. In the equation \(x = -|y|\), the graph exhibits symmetry over the horizontal axis due to the nature of the absolute value function.

• Both \(y = a\) and \(y = -a\) give the same absolute value, \(|y| = \pm a\).
• Consequently, both map to the same \(x\) value: \(x = -a\).

This results in the graph appearing as a vertically symmetric shape, where for each \(x\), positive and negative \(y\) values align perfectly, mirroring each other along the x-axis.
This kind of symmetry directly contributes to the failure of the vertical line test, as discussed earlier. Multiple \(y\) values leading to the same \(x\) value creates a vertical stack of points, confirming that the graph cannot be classified as a function.