/*! 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 12 Determine the domain of each fun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 12

Determine the domain of each function of two variables. $$k(x, y)=\frac{1}{x}+\frac{y}{x-1}$$

Short Answer

Expert verified
The domain is \[ \{(x, y) \ | \ x eq 0, \ x eq 1, \ y \ \text{is any real number} \} \]

Step by step solution

01

Identify restrictions from each term

For the function to be defined, each term must be defined. In the first term, \( \frac{1}{x} \), the denominator cannot be zero, so \( x eq 0 \). In the second term, \( \frac{y}{x-1} \), the denominator cannot be zero, so \( x eq 1 \).
02

Combine the restrictions

Combine the conditions found in Step 1 to determine the domain. The combined condition is that \( x eq 0 \) and \( x eq 1 \).
03

Express the domain in set notation

The domain can be expressed in set notation as: \[ \{(x, y) \ | \ x eq 0, \ x eq 1, \ y \ \text{is any real number} \} \]

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.

functions of two variables
A function of two variables is a rule that takes a pair of numbers and assigns a single value to this pair. These functions are often written as \( f(x, y) \), where \( x \) and \( y \) represent the input variables. Each pair of values for \( x \) and \( y \) corresponds to a unique output value.
domain restrictions
The domain of a function includes all the possible input values for which the function is defined. To find the domain, you need to identify restrictions where the function cannot be evaluated. These restrictions usually occur where a term in the function involves division by zero, or where the expression under a square root is negative. In our example, the term \( \frac{1}{x} \) cannot have \( x = 0 \) because division by zero is undefined. Similarly, the term \( \frac{y}{x-1} \) cannot have \( x = 1 \), as it would also result in a division by zero.
set notation
Set notation is a shorthand used to define the domain of a function. It helps to clearly write down all the conditions that the variables need to satisfy. In the given example, the restrictions are \( x eq 0 \) and \( x eq 1 \), which means \( x \) can take any value except 0 and 1. Additionally, \( y \) can be any real number. Using set notation, we write the domain as: \[ \{ (x, y) \ | \ x eq 0, \ x eq 1, \ y \text{ is any real number} \} \]

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

Describe the geometric meaning of the double integral of a function of two variables.

Find the indicated maximum or minimum values of \(f\) subject to the given constraint. Minimum: \(f(x, y)=x y ; \quad x^{2}+y^{2}=9\)

Find the relative maximum and minimum values and the saddle points. $$f(x, y)=x y+\frac{2}{x}+\frac{4}{y}$$

Find the indicated maximum or minimum values of \(f\) subject to the given constraint. Maximum: \(f(x, y, z, t)=x+y+z+t\) \(x^{2}+y^{2}+z^{2}+t^{2}=1\)

Consider the following data showing the average life expectancy of men in various years. Note that \(x\) represents the actual year. $$\begin{array}{|cc|}\hline & \text { Life Expectancy } \\\\\text { Year, } x & \text { of Men, } y \text { (years) } \\\1950 & 65.6 \\\1960 & 66.6 \\\1970 & 67.1 \\\1980 & 70.0 \\\1990 & 71.8 \\\2000 & 74.1 \\\2003 & 74.8 \\\\\hline\end{array}$$ a) Find the regression line, \(y=m x+b\) b) Use the regression line to predict the life expectancy of men in 2010 and 2015.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.