Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the equation represents \(y\) as a function of \(x.\) $$|y|=3-2 x$$

Step-by-Step Solution

Answer

No, the given equation does not represent \(y\) as a function of \(x\).

Step 1: Isolate Variable

The equation can be rewritten as \(y = \pm (3-2x)\) by applying formula of absolute value, which states that \(|y| = y\) when \(y \geq 0\) and \(|y| = -y\) when \(y < 0\).

Step 2: Check for all possible outputs

The \(\pm\) sign indicates that the equation \(y = \pm (3-2x)\) has two possible solutions, \(y = 3-2x\) and \(y = -(3-2x)\), for each value of \(x\). Therefore, for a particular \(x\), there can be two possible values of \(y\).

Step 3: Determine if y is a function of x

A relationship is a function if for every value of one variable, there is exactly one value of the second variable. But as per the above explanation, the equation does not represent y as a function of x, as there are two potential values of y for each x.

Key Concepts

Understanding Absolute Value

The absolute value of a number is the distance of that number from zero on the number line, regardless of direction. Think of it as the 'numerical value' without considering the sign. For example, both \(3\) and \(\text{-}3\) have the same absolute value, \(3\), because they are each three units away from zero on the number line.

The symbol for absolute value is two vertical bars encompassing a number or expression, like this: \( |y| \) or \( |\text{-}3| \) which equals \( 3 \). The fundamental idea is that an absolute value equation can have two possible solutions, one positive and one negative, due to this distance property. This concept is crucial to grasp because it often leads to a split in possible solutions for variables within absolute value symbols, as seen in the exercise.

Defining a Function

A function is a special type of relation where every input (often represented by the variable \(x\)) corresponds to exactly one output (commonly represented by \(y\)). This is known as the 'vertical line test,' where if you can draw a vertical line through a graph of the relation and it touches the graph in more than one point at any place, it's not a function.

The formal function definition ties together an input set (domain) and an output set (range) in a consistent, predictable manner. Mathematicians often write functions in the form \( f(x) = {expression} \) which reads as 'f of x equals...' where \( f \) is the function name, and the expression defines the relationship. For instance, in a linear function like \( f(x) = 2x + 3 \), for every \( x \), you'll find a single \( y \) value.

Isolating Variables

Isolating a variable means rearranging an equation so that the variable of interest is alone on one side of the equation. This is a fundamental technique in algebra for solving equations. To isolate a variable, you perform operations such as addition, subtraction, multiplication, division, and applying functions inversely - all while performing the same operation on both sides to maintain the equation's balance.

When you encounter an equation with absolute value, this process can become slightly more complex because you may need to consider two scenarios: when the value inside the absolute value is positive and when it's negative. Therefore, isolating variables in absolute value equations could lead to multiple expressions representing a range of possible solutions.