Problem 23 - Thomas Calculus

Question

Graph the following equations and explain why they are not graphs of functions of \(x .\) $$ |y|=x \quad \text { b. } y^{2}=x^{2} $$

Step-by-Step Solution

Answer

Both equations define relations where each x-value corresponds to two y-values, thus they are not functions of x.

Step 1: Understanding Absolute Value

When dealing with the equation \(|y| = x\), remember that it means either \(y = x\) or \(y = -x\). This represents two distinct lines: one with a positive slope (\(y = x\)) and one with a negative slope (\(y = -x\)).

Step 2: Graphing Absolute Value Relationships

The graph of \(|y| = x\) consists of two lines that cross the origin (0,0): one line heads in the positive direction along both the x and y axes, and another heads in the negative direction along the y axis and in the positive direction along the x axis. Note that for every x (except 0), two different y-values exist.

Step 3: Analyzing Function Definition for |y|=x

A relation is a function if every input (x value) has exactly one output (y value). In \(|y| = x\), for any x-value (except zero), there are two corresponding y-values, violating the function definition. Thus, \(|y| = x\) is not a function of x.

Step 4: Interpreting the Squared Equation

The equation \(y^2 = x^2\) suggests that both y and x can take either positive or negative values. Specifically, it breaks down into \(y = x\) and \(y = -x\), similar to the absolute value scenario but with a different approach.

Step 5: Graphing Squared Relationships

Graph \(y^2 = x^2\) by recognizing it as two separate lines defined by \(y = x\) and \(y = -x\). This forms a pair of diagonal lines intersecting at the origin, similar to an 'X' on the graph, with each segment reflecting through the origin.

Step 6: Evaluating Function Criteria for y^2=x^2

Since \(y^2 = x^2\) means for each x-value there are two possible y-values (besides zero), it does not satisfy the definition of a function where a single x-value must map to only one y-value. So, \(y^2 = x^2\) isn't a function of x either.

Key Concepts

Absolute Value

Absolute value represents the distance a number is from zero on a number line, without considering direction. In mathematical terms, it's denoted by vertical bars, like \(|a|\) for a number \(a\). When you have \(|y| = x\), it implies two possible equations: \(y = x\) and \(y = -x\). This is because any positive number is at distance \(x\) from zero in either direction.

• For \(y = x\), as \(x\) increases, \(y\) increases on a positive slope through the origin.
• For \(y = -x\), the graph reflects in the line \(y = -x\) descending through the origin.

In this setup, for every \(x\), except for zero, you get two potential values for \(y\). This breaks the rule of associating each \(x\) with a single, unique \(y\) value in functions.

Squared Equations

Squared equations, like \(y^2 = x^2\), relate to both variables raised to the power of two. It indicates that the absolute values of \(y\) and \(x\) are equal. The equation unfolds as two expressions: \(y = x\) and \(y = -x\). This setup is similar to the absolute value equation but highlights squared terms.

• This results in two linear graphs intersecting at the origin.
• These graphs appear as diagonal lines, one moving upwards ( extbackslash( y = x extbackslash)) and the other downwards ( extbackslash( y = -x extbackslash)).

Because squaring operation eliminates the sign, for each \(x\), you come out with two possible \(y\) values other than zero. This mismatches with a function's requirement for each \(x\) to map to exactly one \(y\).

Graphing Techniques

Graphing is a crucial technique in visualizing mathematical relationships. To graph \(|y| = x\) and \(y^2 = x^2\), begin by identifying their line equations. Both show two lines that cross at the origin.

• For \(|y| = x\), draw two lines: \(y = x\) and \(y = -x\), meeting at (0,0).
• For \(y^2 = x^2\), follow the same: draw lines \(y = x\) and \(y = -x\).

Both result in an 'X' shape. This graphing unveils how, visually, the equations have multiple \(y\) values for a single \(x\) input beyond zero, clarifying their status as non-functions.

Function Definition

A function has a strict definition in mathematics: each input (or \(x\) value) must map to exactly one output (or \(y\) value). This crucial criterion differentiates functions from non-functions.

• An equation passes the 'vertical line test' if for any drawn vertical line, it meets the graph at most once.
• For equations like \(|y| = x\) and \(y^2 = x^2\), a vertical line will touch them twice, confirming they are not functions.

Each equation fails the function definition by allowing two possible values for \(y\) when \(x\) isn't zero.

Intersecting Lines

The concept of intersecting lines is fundamental in understanding relationships between equations graphically. When we graph the lines from \(|y| = x\) and \(y^2 = x^2\), the intersection occurs at the origin (0,0).

• The intersecting lines create an 'X' formation on the graph.
• The intersection demonstrates multiple intersecting inputs, reinforcing the non-function nature of these equations.

Through graphing intersecting lines, we see why certain equations fail the function criterion as they provide more than one output for a given input.