/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 Determine whether each relation ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 63

Determine whether each relation defines y as a function of \(x .\) (Solve for y first if necessary.) Give the domain. $$ y=-\frac{2}{x} $$

Short Answer

Expert verified
Yes, y is a function of x. The domain is \( (-\infty, 0) \cup (0, +\infty) \).

Step by step solution

01

Understand the Definition of a Function

A function is a relation where each input (x) has exactly one output (y). To determine if the given equation defines y as a function of x, check if for each x-value there is exactly one corresponding y-value.
02

Solve for y

The given equation is already solved for y: \( y = -\frac{2}{x} \)
03

Determine If It Is a Function

Analyze if there is exactly one y-value for each x-value. In this equation, for any non-zero x, there is exactly one corresponding y-value. Therefore, y is a function of x.
04

Determine the Domain

The domain of a function is the set of all possible input values (x-values) that will output a real number. Since division by zero is undefined, x cannot be zero in this equation. Therefore, the domain is all real numbers except zero.In set notation, the domain is: \( (-\infty, 0) \cup (0, +\infty) \)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Definition of a Function
To determine whether a relation defines y as a function of x, we must understand what a function is. In algebra, a **function** is a special relationship between x (input) and y (output), where each input value (x) has exactly one corresponding output value (y).
This means if we plug any x-value into our relation, we should get one and only one y-value. If we can find any x-value that gives us multiple y-values, then our relation is not a function.
In the given exercise, the relation is represented by the equation \( y = -\frac{2}{x} \). To identify whether this is a function, we need to ensure that every x-value maps to only one y-value.
Solving for y
Sometimes, relations can be given in a form where y is not isolated. In such cases, solving for y means rearranging the equation to express y explicitly in terms of x.
In the given problem, the expression is already solved for y: \( y = -\frac{2}{x} \). This means y is written as a function of x directly.
Whenever you're asked to determine if a relation is a function, one of the first steps is to isolate y if it鈥檚 not already. This makes it easier to analyze the relationship between x and y values.
Domain of a Function
The **domain** of a function is the set of all possible x-values (inputs) that can be plugged into the function without causing undefined or non-real results.
For the given function \( y = -\frac{2}{x} \), we need to determine all x-values that make the expression valid.
In this case, for \( y = -\frac{2}{x} \), x cannot be zero because division by zero is undefined. Therefore, the function is defined for all real numbers except x=0.
In interval notation, the domain is written as: \( (-\infty, 0) \cup (0, +\infty) \). This tells us that x can be any real number except zero.
Input and Output Values
In any function, **input values** (x-values) are the values you can substitute into the function, while **output values** (y-values) are the values you get after performing the function's operation on input values.
For the function \( y = -\frac{2}{x} \), the possible outputs depend directly on the inputs.
For example, if you input \( x = 2 \), then \( y = -\frac{2}{2} = -1 \). Here, 2 is the input, and -1 is the output. Conversely, if you input \( x = -2 \), then \( y = -\frac{2}{-2} = 1 \).
Because each x-value except zero has exactly one corresponding y-value, this confirms that \( y = -\frac{2}{x} \) is indeed a function of x.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ \left(\frac{h}{g}\right)(x) $$

For \(f(x)=2 x+3\) and \(g(x)=x+5,\) a student found \((f \circ g)(x)\) correctly as $$ (f \circ g)(x)=2 x+13 $$

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ (g+h)\left(\frac{1}{3}\right) $$

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (h \circ f)(x) $$

Determine whether each relation defines y as a function of \(x .\) (Solve for y first if necessary.) Give the domain. $$ y=x^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.