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Magazine Chapter 3: Problem 56
Exer. 55-56: Explain why the graph of the equation is not the graph of a function. $$ x=-|y| $$
Short Answer
Expert verified
The graph fails the vertical line test; thus, it is not a function.
Step by step solution
01
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 \( y \).
02
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 \( x \).
03
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 \( \leq 0 \).
04
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.
05
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.
06
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 \( y \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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: \(-|y| \leq 0\).
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.
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.
- 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 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.
