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Chapter 2: Problem 63

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

Short Answer

Expert verified
The equation does not define \(y\) as a function of \(x\).

Step by step solution

01

Understand the Problem

We need to determine if the equation \(x = y^2\) defines \(y\) as a function of \(x\). A function means every \(x\) has a unique \(y\) value.
02

Solve for y

Attempt to express \(y\) explicitly in terms of \(x\). Start by solving the equation \(x = y^2\) for \(y\).
03

Solve the Equation

Rearrange the equation \(x = y^2\) to solve for \(y\):\[y = \pm \sqrt{x}\]
04

Determine Uniqueness

Examine the solution \(y = \pm \sqrt{x}\). There are two possible values for \(y\) (one positive and one negative) for each positive \(x\), implying \(y\) is not uniquely determined by \(x\).
05

Conclusion

Since there are two possible \(y\) values for most \(x\) (except \(x = 0\) where \(y = 0\)), \(y\) is not a function of \(x\).

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Key Concepts

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

Uniqueness in Functions
To determine if an equation like \(x = y^2\) defines a function, we must understand the concept of uniqueness in functions. In mathematical terms, a function is a relationship between two sets, specifically an input set (domain) and an output set (codomain), where each input is related to exactly one output.
When we scrutinize the equation \(x = y^2\), we are examining if every value of \(x\) correlates with a single, unique value of \(y\).
Let鈥檚 break down the possible solutions to see if they comply with this requirement for uniqueness.
  • If the equation provides a unique solution for \(y\) in terms of \(x\), it defines a true function.
  • If there are multiple solutions for \(y\), then the equation does not qualify as a function.
This uniqueness condition is critical in defining whether a relationship can be classified as a function.
Solving Equations for Function Definition
When tasked with the equation \(x = y^2\) and determining if it defines a function, we first need to solve it for \(y\). This process involves rearranging the equation so that \(y\) is on one side.
Looking at the given equation:
  • Start with \(x = y^2\).
  • To solve for \(y\), take the square root of both sides, leading to: \(y = \pm \sqrt{x}\).
Here, the sign \(\pm\) indicates two potential solutions for \(y\): one positive and one negative.
This means for any positive \(x\), \(y\) could be either \(\sqrt{x}\) or \(-\sqrt{x}\).
At this point, it becomes clear that there isn't just one solution, which relates directly to the concept of uniqueness in function definition. Thus, solving equations is a crucial step in investigating the possibility of a functional relationship.
Explicit Form in Function Representation
An explicit form of an equation is one where the dependent variable, typically \(y\), is expressed solely in terms of the independent variable, \(x\).
In our equation \(x = y^2\), solving explicitly means writing \(y\) as a direct formula with respect to \(x\).
After rearranging, our solutions are \(y = \pm \sqrt{x}\).
This formulation indicates a problem with defining \(y\) as a function: it is not presented in a unique explicit form.
  • An explicit function form would imply a single expression of \(y\), such as \(y = f(x)\).
  • Here, the expression \(y = \pm \sqrt{x}\) fails the explicit uniqueness test as it gives two possible \(y\)-values for each \(x\), other than zero.
Achieving an explicit unique form is fundamental in confirming a true function relationship.

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Most popular questions from this chapter

Sums of Even and Odd Functions If \(f\) and \(g\) are both even functions, is \(f+g\) necessarily even? If both are odd, is their sum necessarily odd? What can you say about the sum if one is odd and one is even? In each case, prove your answer.

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