Problem 10 - College Algebra with Corequisite Support

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.

Step 1: Understand the Definition of a Function

A relation represents \(y\) as a function of \(x\) if for each input value \(x\), there is exactly one output value \(y\). This means that each \(x\) must map to only one \(y\).

Step 2: Analyze the Given Equation

We are given the equation \(x = y^2\). To determine if this is a function, we need to consider how values of \(x\) relate to values of \(y\).

Step 3: Isolate y to Examine y in terms of x

To examine \(y\) in terms of \(x\), solve for \(y\) as a function of \(x\). Start by rewriting the equation: \(y = \pm \sqrt{x}\).

Step 4: Determine if Each x Leads to One y Value

The equation \(y = \pm \sqrt{x}\) indicates that for each non-negative \(x\), there are two potential values for \(y\): \(\sqrt{x}\) and \(-\sqrt{x}\). This means \(x = y^2\) does not satisfy the definition of a function since each \(x\) has more than one possible \(y\) value.

Key Concepts

Definition of a Function

In mathematics, a function is a specific type of relation. A relation simply describes how two sets of values relate to each other. However, for a relation to be a function, it must satisfy a key criterion: for every input value (often called 'x'), there must be exactly one corresponding output value ('y').

Think of it like a vending machine: for every button you press (the input), you should get a single product (the output). If you sometimes get two different products for one button, it wouldn't work correctly as a vending machine, just as it wouldn't qualify as a function in mathematics.

Thus, the definition of a function restricts each input to yield exactly one output. This is the main principle we need to keep in mind.

Relation Analysis

Analyzing relations involves taking a close look at how values from one set correspond to values in another set. In our exercise, we are given the relation in the form of an equation: \( x = y^2 \).

To understand this relation, we should ask whether each value of \( x \) uniquely determines a value of \( y \). Analyzing involves tracing how setting \( x \) to a specific number maps to potential \( y \/ s \), which involves considering the equation structure itself.

This step is crucial because identifying whether a relation is indeed a function helps us understand mathematical connections and lays the groundwork for deeper studies in calculus and algebra.

Equation Solving

Equation solving involves manipulating the mathematical statement to find values or express one variable in terms of another. To determine whether the relation \( x = y^2 \) is a function, we rearrange the equation to express \( y \) in terms of \( x \).

Through solving, we get two equations: \( y = \sqrt{x} \) and \( y = -\sqrt{x} \). These solutions tell us that for each \( x \) (as long as \( x \geq 0 \)), there are two corresponding \( y \) values. This outcome is unconventional for a function, as a function requires only one output for each input.

Equation solving successfully shows how variables link but also highlights the need to follow the function definition rigorously.

Mapping of Variables

Mapping of variables illustrates exactly how input values are paired with output values in a relation, which is crucial for understanding if a relationship is a function.

With our given equation, \( x = y^2 \), mapping means examining how every potential \( x \) value results in multiple relationships with \( y \). Upon solving, for example, if \( x = 4 \), then \( y \) could be \( +2 \) or \( -2 \).

This mapping shows every positive \( x \) relates to two \( y \) values, reflecting a one-to-many relationship rather than one-to-one mapping expected of a function. Mapping helps confirm, through visual or logical assessment, that a relation doesn't adhere to function criteria. Instead of representing a precise, single output for every input, it demonstrates how some equations create several options, violating the function condition.