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Chapter 2: Problem 56

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

Short Answer

Expert verified
The equation is not a function because each \(x < 0\) corresponds to two \(y\) values, failing the vertical line test.

Step by step solution

01

Identify the Expression

The given equation is \(x = -|y|\) where \(|y|\) represents the absolute value of \(y\). The absolute value function \(|y|\) outputs the non-negative version of \(y\) regardless of its sign.
02

Solve for y

To understand why this is not a function, consider the relationship implied: \(x = -|y|\). This expression can be rearranged to express \(y\): \(-|y| = x\), which implies \(|y| = -x\). Since absolute values are always non-negative, this is only possible if \(x\) is non-positive (\(x \leq 0\)).
03

Examine Multiple Outputs

The condition \(|y| = -x\) means that \(y\) can be any of the values \(y = -x\) or \(y = x\). For a single \(x\) value, there can be two possible \(y\) values: for instance, if \(x = -4\), then \(y = 4\) or \(y = -4\).
04

Check the Vertical Line Test

The vertical line test states that if a vertical line intersects a graph in more than one point, the graph does not represent a function. The equation \(x = -|y|\) fails this test because, at any point \(x < 0\), multiple \(y\) values correspond to a single \(x\) value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Function
Absolute value functions are a fundamental concept in mathematics. The absolute value of a number measures its magnitude without regard to its sign.
In simpler terms, the absolute value \(|y|\) of a number \(y\) is always non-negative, whether \(y\) itself is positive or negative.
  • The absolute value of a positive number is the number itself: \(|3|=3\).
  • The absolute value of a negative number is the number made positive: \(|-3|=3\).
Understanding this property is essential for interpreting equations like \(x = -|y|\). This specific equation involves the absolute value function but rearranges it to find some relationship between \(|y|\) and \(x\).
Multiple Outputs
In mathematical terms, a function is a relationship where each input corresponds to exactly one output. However, the equation \(x = -|y|\) presents more than one possible output for certain inputs. This is what we refer to as multiple outputs.
  • For each non-positive \(x\) (meaning any \(x\) value where \(x \leq 0\)), two different \(y\) values satisfy the equation \(|y| = -x\).
  • For instance, when \(x = -4\), the values of \(y\) could be either \(4\) or \(-4\), as both satisfy \(|y|=4\).
This characteristic signifies that the graph described by the equation \(x = -|y|\) does not represent a mathematical function due to these multiple potential outputs for single \(x\) inputs.
Function Graph
The graphical representation of an equation like \(x = -|y|\) provides visual insight into the relationship between \(x\) and \(y\). In a typical function graph, each value of \(x\) should intersect with the graph at only one point, which represents the single output for the function's input.
However, the graph of \(x = -|y|\) is different:
  • Vertical lines drawn at \(x < 0\) intersect the graph at two points, corresponding to the two \(y\)-values for each \(x\) input.
  • This violates the "vertical line test" for functions, a method used to determine if a graph represents a function.
When a vertical line crosses the graph more than once, it highlights that the equation is not a function. The vertical line test is a simple yet effective tool for verifying function qualities on a graph.
Non-Positive x-values
An important detail about the equation \(x = -|y|\) is that it inherently restricts \(x\) to non-positive values. This means that for the equation to be valid, \(x\) must always be less than or equal to zero.
  • This is because the expression \(|y| = -x\) must hold, and \(|y|\) is always non-negative or zero.
  • If \(x\) were positive, \(y\) could not have an absolute value equating to a negative \(x\) value.
In essence, the nature of absolute values combined with the negative sign in \(-|y|\) automatically limits \(x\) to zero or negative numbers, dictating the possible inputs for this non-function relationship.
This insight into non-positive x-values is crucial for understanding why the graph of \(x = -|y|\) cannot be a function on a standard \(xy\) coordinate plane.

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Most popular questions from this chapter

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Sketch the graph of the equation. $$y=|| x|-1|$$
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