Problem 16 - Beginning and Intermediate Algebra

Question

Determine whether each relation describes \(y\) as a function of \(x\) $$y=|x|$$

Step-by-Step Solution

Answer

The relation \(y = |x|\) meets the definition of a function, as it assigns exactly one output (y) to each input (x) with no multiple outputs per input or unassigned outputs. Hence, the relation does describe y as a function of x.

Step 1: To determine whether the relation \(y=|x|\) describes a function, we need to analyze the properties of the absolute value. The absolute value of a number is the distance between the number and zero on the number line, which can be represented as: \( |x| = \begin{cases} x & \text{if}\ x \geq 0 \\ -x & \text{if}\ x < 0 \end{cases} \) #Step 2: Test for unassigned outputs#

Using the definitions from step 1, we can see that for any positive or negative value of x, there will always be a corresponding value of y: - If x ≥ 0, y = x - If x < 0, y = -x Since there is always a corresponding value of y for any value of x, there are no unassigned outputs. #Step 3: Test for multiple outputs per input#

Step 2: The relation \(y=|x|\) assigns exactly one output to each input: - If x ≥ 0, y = x (this rule assigns only one value of y) - If x < 0, y = -x (this rule also assigns only one value of y) Since there is always exactly one value of y for any value of x, there are no multiple outputs per input. #Step 4: Conclusion#

Since there are no unassigned outputs and there are no multiple outputs per input, the relation \(y=|x|\) meets the definition of a function. Therefore, the relation does describe y as a function of x.

Key Concepts

Absolute Value

The absolute value of a number can be understood as its distance from zero on the number line, without considering whether it's positive or negative. In mathematics, absolute value is denoted by two vertical bars surrounding the number, like this: \(|x|\).
For example, both 5 and -5 have an absolute value of 5 because they are both five units away from zero.

• If the number is positive or zero, the absolute value is the number itself (e.g. \(|5| = 5\)).
• If the number is negative, the absolute value is the positive version of that number (e.g. \(|-5| = 5\)).

Using absolute value in a function, such as \(|x|\), helps us determine values in a consistent manner regardless of the sign of the input number. This is crucial for analyzing mathematical relationships and properties.

Relation

In mathematics, a relation describes how two or more quantities or elements relate to each other. With functions, you're determining if one variable is related to another in a specific way. For example, the equation \(y = |x|\) forms a relation between \(x\) and \(y\).
Relations can be visualized by plotting them on graphs, representing the relationship between axes or dimensions. \(y = |x|\) forms a V-shape when graphed, showcasing that as \(x\) changes, \(y\) remains non-negative and increases or decreases maintaining its absolute nature.

• In this relation, no matter what value \(x\) is, positive or negative, \(y\) remains positive (or zero).
• This one-to-one correspondence without multiple values of \(y\) for each \(x\) ensures it meets the criteria of a function.

Output

In mathematical functions, the output is what you get after putting an input through the function. For example, in the function \(y = |x|\), \(y\) is the output resulting from any value you substitute in for \(x\) (the input).
The importance of having one output for each input ensures consistency and reliability in predictions based on the function:

• For \(x = 3\), the output \(y = |3| = 3\).
• For \(x = -3\), the output \(y = |-3| = 3\).

The function evaluates values by converting negative numbers to their positive counterparts, illustrating that the outputs maintain a clear relationship to the inputs through their absolute percentage.

Input

The input is the variable or number you introduce into a function to see what output it generates. For the function \(y = |x|\), \(x\) is the input.
This involves testing different input values to comprehend how the function behaves or processes those values:

• Always substitute inputs into \(x\) to gain the exact outcome in \(y\).
• Understand that whether \(x\) is positive or negative, it yields a predictable and single positive value in \(y\).

By clearly distinguishing the input, we ensure that each is mapped precisely, leading to a function's credibility in its domain. This assurance of singular output strengthens the function's integrity and utility in mathematical problem-solving.