Problem 25 Determine whether the equation d... [FREE SOLUTION]

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Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 3: Problem 25

Determine whether the equation defines $y$ to be a function of $x$. $$x=-y^{2}$$

Short Answer

Expert verified

The equation $x=-y^{2}$ does not define y as a function of x. Since every value of x corresponds to two values of y, the definition of a function is contradicted.

Step by step solution

Express the Equation in the form $y = f(x)$

Firstly, let's rearrange the equation into $y = f(x)$ form. Solving $x=-y^{2}$ for y, we have $y = \sqrt{-x}$ and $y = -\sqrt{-x}$. The positive or negative square root of x are two potential values of y for the given x.

Determine Whether Each Value of x Corresponds to a Single Value of y

Look at the expressions $y = \sqrt{-x}$ and $y = -\sqrt{-x}$. They show that for any value of x, there are two possible values of y, a positive and a negative. This contradicts the definition of a function, which states that each x-value should correspond to exactly one y-value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function of a Variable

A function of a variable is a relationship between two sets, typically expressed as a rule that assigns to every element from one set (input) a single element from another set (output). When we talk about functions involving variables, we often denote these sets as the domain (for inputs) and the range (for outputs). For example, if you have a function $ y = f(x) $, $ x $ is the independent variable (input) and $ y $ is the dependent variable (output).
The essence of a function is its predictability and consistency; for each input, there's always a corresponding output that stays the same when you check repeatedly. This reliability is key in mathematics and its applications.

Relation between Variables

Understanding the relation between variables is crucial as it helps us decode complex systems and analyses. In any equation involving variables, such as our given equation $ x = -y^2 $, the variables $ x $ and $ y $ show a relationship.

The equation suggests how changes in one variable might affect the other.
In this context, $ y $ depends on $ x $, implying a directional relationship.
Variables can share direct, inverse, or more complex relationships.

In our example, rearranging it into forms $ y = \sqrt{-x} $ and $ y = -\sqrt{-x} $ tells us that with every possible $ x $, $ y $ could potentially have two values, showing a complex relation rather than a straightforward one-to-one map that characterizes functions.

Function Definition

The definition of a function is fundamental in mathematics. A function can be loosely defined as a rule that links each value of one variable directly to precisely one value of another. The primary idea is about this unique correspondence.

A valid function from $ x $ to $ y $ ensures that every $ x $ value (an element in its domain) maps to one and only one $ y $ value.
This idea is central for function analysis, as it preserves mathematical consistency and reliability.
If an equation like $ x = -y^2 $ allows more than one output for any single input, it breaks the rule that defines a function.

Thus, for an equation to truly be a function, it needs to uphold this one-to-one correspondence rule. This is why, in our example, $ x = -y^2 $ cannot generally be classified as a function.

Solving Equations

Solving equations is a fundamental skill in algebra and calculus that requires finding values of variables that make the equation true. It involves various techniques and can serve numerous purposes depending on the equation type.

For equations to define functions, the typical goal is to solve for dependent variables explicitly in terms of independent ones.
With $ x = -y^2 $, solving for $ y $ involves understanding the implications of $ y = \pm\sqrt{-x} $.
This solution reflects that at any specific $ x $, there are potentially two $ y $ solutions, showing a complication in defining a straightforward function.

Solutions of these sorts of equations help identify whether one input precisely determines a single output, aligning or conflicting with functional relationships as defined in math.

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91影视

Determine whether the equation defines \(y\) to be a function of \(x\). $$x=-y^{2}$$

Short Answer

Step by step solution

Express the Equation in the form \(y = f(x)\)

Determine Whether Each Value of x Corresponds to a Single Value of y

Key Concepts

Function of a Variable

Relation between Variables

Function Definition

Solving Equations

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