Intermediate Algebra, Chapter 3, Review Exercises, Section Review Exercises, Problem 48

Determine whether the equation $y = |x|$ defines $y$ as a function of $x$. Give the domain in each case. Identify any linear functions.


By using the property of Absolute value, we have

$y = |x| \Longrightarrow y = \begin{array}{c} x & \text{for} & x > 0 \\ \\ -x & \text{for} & x < 0 \end{array}$


The graph of this function is obtained by the union of the line $y = x$ and $y = -x$.
And because those lines are not vertical lines, we can say that the given relation defines
a function because there is only one corresponding value of $y$ for every value of $x$.
However, the factor is defined for every values of $x$ except for . Therefore, the domain
is $(-\infty, 0) \bigcup (0, \infty)$. This equation is not an example of a linear equation
because the slope is not constant all throughout the graph.