Problem 10 - College Algebra

Question

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). $$ x=y^{2} $$

Step-by-Step Solution

Answer

The relation does not represent \(y\) as a function of \(x\) because \(x = y^2\) results in two possible \(y\) values for one \(x\).

Step 1: Understanding the Relation

The given relation is \(x = y^2\). Here, \(x\) is written in terms of \(y\). To check if this relation represents \(y\) as a function of \(x\), we need to understand how \(y\) behaves for each \(x\).

Step 2: Analyzing the Equation

For any given \(x \geq 0\), the equation \(x = y^2\) can have two solutions for \(y\), which are \(y = \sqrt{x}\) and \(y = -\sqrt{x}\). This means \(y\) can have two values for one \(x\) when \(x\) is non-negative.

Step 3: Determine if Each Input Has Exactly One Output

For \(y\) to be a function of \(x\), there should be exactly one \(y\) value for each \(x\) value. Since for \(x = 4\), \(y\) can be \(2\) or \(-2\), \(y\) does not have a unique output for every \(x\). Therefore, this relation is not a function.

Key Concepts

Relation

A relation in mathematics is a connection between two sets of values, where each element of one set is paired with one or more elements of another set. In the given exercise, the relation is expressed by the equation \(x = y^2\). Here, values of \(x\) are related to values of \(y\). The challenge is to identify if this relation can act as a function.To better understand relations:

• A relation can exist between numbers, shapes, or even colors.
• Relations are often represented using ordered pairs \((x, y)\).
• A relation displays how elements from one set are associated with elements from another set.

When you analyze relations, try to see if they represent functions. Functions are special kinds of relations. Let's explore more about this!

One-to-One Correspondence

One-to-one correspondence is a specific type of function where each input has a unique output, and each output has a single input. However, this contrast sharply with the relation \(x = y^2\) given in the exercise.In our relation:

• The equation \(x = y^2\) shows that for a given \(x\), there might be multiple \(y\) values, specifically, \(y = \sqrt{x}\) and \(y = -\sqrt{x}\).
• This means that \(x = 4\) could correspond to both \(y = 2\) and \(y = -2\). Hence, it fails to be a one-to-one correspondence.

One-to-one correspondence ensures no overlap or duplication between inputs and outputs. If a relation does not have a one-to-one correspondence, it cannot be classified as a function.

Function of a Variable

A function of a variable is a relation that assigns exactly one output for every input. For \(y\) to be a function of \(x\), each outcome in the set of \(x\) values must map to only one \(y\) value.Consider the equation \(x = y^2\):

• For this equation to depict \(y\) as a function of \(x\), every value of \(x\) would need to relate to exactly one \(y\) value.
• However, as analyzed, values like \(x = 4\) lead to two different \(y\) values (\(2\) and \(-2\)).

Therefore, the relation is not a function because the uniqueness condition for function mapping is violated. Functions are essential in mathematical analysis, providing predictable and consistent relationships.

Mathematical Analysis

Mathematical analysis involves studying functions and their properties, and understanding their underlying principles.In examining this relation \(x = y^2\), mathematical analysis helps us:

• Break down how the elements of the relation relate to each other.
• Conclude that the relation cannot be a function due to the multiple \(y\) outputs for a single \(x\) input.

Through analysis, you learn to identify patterns and behaviors in mathematical expressions. Here, constraints like one-to-one mapping become evident. Mathematical analysis equips you with tools to decipher complex relationships, preparing you for deeper mathematical explorations.