Problem 28 - Algebra A Combined Function

Question

Determine whether each relation is also a function. \(x=y^{2}+2\)

Step-by-Step Solution

Answer

The relation is not a function because an x-value corresponds to multiple y-values.

Step 1: Understand the Definition of a Function

A relation is a function if each input (x-value) corresponds to exactly one output (y-value). In other words, for each value of y, there must be only one x-value.

Step 2: Identify Variables and Equation Format

The given relation is \( x = y^2 + 2 \). Here, \( y \) is the input (independent variable) and \( x \) is the output (dependent variable).

Step 3: Analyze the Relation

We need to consider if for each \( y \), the output \( x \) is unique. We observe that if \( y_1 = y_2 = 3 \) and \( y_1 = -y_2 = 3 \) in the equation will result in \( x = 11 \) because \( x = (3)^2 + 2 = (-3)^2 + 2\).

Step 4: Evaluate the Function Criterion

Notice that for a single value of \( x \), there can be multiple values of \( y \). For instance, \( x = 4 \) corresponds to \( y = \pm \sqrt{2} \). Therefore, a single value of \( x \) does not define a single value of \( y \), which violates the criterion for the function.

Key Concepts

Independent and Dependent Variables

In mathematics, understanding independent and dependent variables is crucial for analyzing relationships between quantities. In any given equation, these variables play specific roles.

• **Independent Variable**: This is the variable you can manipulate freely. In the given problem, this is represented by the variable \( y \). You choose any value for \( y \), and it will influence the output.
• **Dependent Variable**: This is the variable that depends on the value of the independent variable. Here, \( x \) is the dependent variable because it changes as \( y \) varies. Specifically, \( x \) is calculated using the formula \( x = y^2 + 2 \).

The role of these variables helps determine which one affects the other, providing a clear way to understand their parts in the relationship.

Function Criterion

A primary criterion for determining if a relation is a function involves observing how each input is related to the output.

• For a relation to be classified as a function, each input value must correspond to exactly one output value.
• In our example, the equation \( x = y^2 + 2 \) does not satisfy this condition because multiple \( y \) values can map to the same \( x \) value.

To assess if something is a function, we must ensure that no single output value points to multiple inputs. A failure in this rule means the "function criterion" is violated, which happens here due to points like \( x = 11 \) having two valid \( y \) values of 3 and -3.

Input-Output Relationship

Understanding the input-output relationship is key to comprehending how inputs (independent variables) influence outputs (dependent variables). This relationship acts as a guideline for evaluating if a set of ordered pairs qualifies as a function.

• The relation \( x = y^2 + 2 \) demonstrates how changing \( y \) affects the resulting \( x \).
• If an equation allows more than one input to map to the same output, like having both 3 and -3 for \( y \) resulting in the same \( x \), the function criterion doesn't hold.

When analyzing such equations, checking the uniqueness of the output for each given input is pivotal for maintaining a valid input-output relationship that fits within the definition of a function. If multiple inputs correspond to one output, it signals the relation isn't a function.