Problem 26 - College Algebra Essentials

Question

Determine whether each equation defines \(y\) as a function of \(x .\) $$ |x|-y=5 $$

Step-by-Step Solution

Answer

No, the equation \( |x| - y = 5 \) does not define \(y\) as a function of \(x\).

Step 1: Analyze the form of the equation

The equation given is \( |x| - y = 5 \). The absolute value of \(x\) alienates the simple linearity of the equation, thus adding an extra layer of complexity.

Step 2: Analyze the effect of the absolute value

The absolute value of \(x\) is always nonnegative (\( |x| >= 0\)), thus for a given \(x\), it will produce two values, one positive and one negative. This suggests that for a given \(x\), there could be two possible \(y\)'s.

Step 3: Reorganize the equation

To visualize this more clearly, reorganize the equation as \( y = |x| - 5 \). For a given \(x\) value, it gives a clear visual that there could be two \(y\)'s. For instance, if \(x=6\), then \(y\) could be \(1\) or \(-1\). This destroys the function.

Key Concepts

Absolute Value

The absolute value of a number can be thought of as its distance from zero on a number line, disregarding whether it is positive or negative. In mathematical terms, it is defined as:

\begin{align*}|x| = \begin{cases} x, & \text{if} x\geq 0 \ -x, & \text{if} x<0\frac{-x}{-1}, & \text{if} x<0\text{-|x}; & \text{if} x<0\end{cases}\end{align*}
This property ensures that the absolute value output is always nonnegative. When dealing with absolute value in equations, it introduces additional considerations because the expression within the absolute value can have two potential outcomes. For example, if an equation includes an absolute value of a variable—such as in our case, \( |x| \)—it means that both \( x \) and \( -x \) must be accounted for when solving for \( y \). The presence of an absolute value in an algebraic expression signifies that there can be more than one solution for a given value of the variable, which directly affects the analysis of functions and their graphs.

Defining Functions

A function in algebra is a relation between a set of inputs and a set of permissible outputs where each input is related to exactly one output. The notation \( f(x) \) is commonly used to denote a function named \(f\) with \(x\) being the input variable. When we talk about defining functions, we are determining the rule or expression that relates inputs to outputs.

For a relationship to be a function, each input \(x\) should correspond to no more than one output \(y\). If an equation like \( |x| - y = 5 \) potentially assigns two different values of \(y\) to a single value of \(x\), then it does not define \(y\) as a function of \(x\). Instead, it is a relation where the input (\(x\)) does not have a unique output (\(y\)). When defining functions, it is critical to ensure that this one-to-one relationship between inputs and outputs is preserved, otherwise the definition of a function is violated.

Analyzing Equations

Analyzing equations is a crucial skill in algebra that involves studying the form, behavior, and implications of algebraic expressions. When presented with an equation like \( |x| - y = 5 \), the objective is to explore how \(y\) behaves in response to varying values of \(x\).

One begins by manipulating the equation to isolate the variable of interest, in this case, \(y\). By reorganizing the equation as \( y = |x| - 5 \), it becomes easier to see the relationship between \(x\) and \(y\). However, due to the absolute value, there are two scenarios to consider for each \(x\): when \(x\) is positive and when it is negative.

• For \( x \) positive: \( y = x - 5 \)
• For \( x \) negative: \( y = -x - 5 \)

This dual possibility indicates that a single \(x\) value can yield two different \(y\) outcomes, challenging the function's rule of assigning exactly one output to each input. In-depth equation analysis often involves examining these types of peculiarities, understanding function definitions, and recognizing the implications of operations like the absolute value on the solutions set.