/*! 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 9 Determine the domain of each fun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 9

Determine the domain of each function of two variables. $$f(x, y)=\sqrt{y-3 x}$$

Short Answer

Expert verified
The domain is \(\{(x, y) \in \mathbb{R}^2 \mid y \geq 3x\}\).

Step by step solution

01

Identify the function's expression

The function given is \(f(x, y) = \sqrt{y - 3x}\). In this function, the operation to focus on is the square root.
02

Consider the square root function

Recall that the square root function is only defined for non-negative arguments. Therefore, for the function \(f(x, y) = \sqrt{y - 3x}\), the expression inside the square root, \(y - 3x\), must be greater than or equal to zero.
03

Set up the inequality

To find where the function is defined, set up the inequality: \[y - 3x \geq 0\]
04

Solve the inequality for y

Rearrange the inequality to solve for \(y\): \[y \geq 3x\]
05

Write the domain

The domain is the set of all \((x, y)\) pairs such that \(y \geq 3x\). Therefore, the domain in set notation is: \[D = \{(x, y) \in \mathbb{R}^2 \mid y \geq 3x\}\]

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
Functions of two variables can be more challenging than single-variable functions because they involve two inputs. In our exercise, the function is given by: \[ f(x, y) = \sqrt{y - 3x} \]The domain of a function of two variables consists of all the possible pairs \((x, y)\) that make the function valid.
To understand this better:
  • The first variable, \(x\), and the second variable, \(y\), can represent different quantities.
  • In our case, we look at pairs \((x, y)\) that we plug into the function to make sense of it.
For functions involving two variables, the domain is often represented visually as a region in the xy-plane.
square root function
The square root function is a common mathematical function, but it has specific rules. It is denoted by \(\sqrt{a}\), where \(a\) represents the number under the square root.
Key properties to remember:
  • The function \( \sqrt{a} \) is only defined when \(a\) is greater than or equal to 0.
  • If \(a\) is negative, the square root is not a real number.
In the context of the given exercise: \[ f(x, y) = \sqrt{y - 3x} \]This means that the expression inside the square root, \(y - 3x\) must be non-negative, i.e., \(y - 3x \geq 0\).
If we solve this inequality, we find that \(y \geq 3x\).
Thus, for the values of \(x\) and \(y\) that satisfy this condition, our square root function will yield real numbers.
solving inequalities
Solving inequalities is an essential skill in algebra and it enables us to find the range of input values that make an expression true. Here we consider the inequality from our exercise: \[ y - 3x \geq 0 \]To solve this inequality:
  • First, rearrange the inequality to isolate \(y\): \[ y \geq 3x \]
  • This inequality tells us that for every \(x\), \(y\) must be at least three times \(x\).
  • Graphically, this means that the points \((x, y)\) lie on or above the line \(y = 3x\).
The solution to the inequality provides the domain where our function \(f(x, y)\) is defined. In mathematical terms, the domain is:
\[ D = \{ (x, y) \in \mathbb{R}^2 \mid y \geq 3x \} \]
This set notation indicates all pairs \((x, y)\) in the real number plane where \(y\) is greater than or equal to \(3x\).

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

A professor wants to predict students' final examination scores on the basis of their midterm test scores. An equation was determined on the basis of data on the scores of three students who took the same course with the same instructor the previous semester (see the following table). $$\begin{array}{|cc|}\hline \text { Midterm } & \text { Final Exam } \\\\\text { Score, } x & \text { Score, } y \\\\\hline 70 \% & 75 \% \\\60 & 62 \\\85 & 89 \\\\\hline\end{array}$$ a) Find the regression line, \(y=m x+b .\) (Hint: The \(y\) -deviations are \(70 m+b-75,60 m+b-62,\) and so on. \()\) b) The midterm score of a student was \(81 \% .\) Use the regression line to predict the student's final exam score.

A triple iterated integral such as $$\int_{1}^{4} \int_{1}^{x^{\prime}} f(x, y, z) d x d y d z$$ is evaluated in much the same way as a double iterated integral. We first evaluate the inside \(x\) -integral, treating \(y\) and z as constants. Then we evaluate the middle \(y\) -integral, treating z as a constant. Finally, we evaluate the outside z-integral. Evaluate these triple integrals. $$\int_{0}^{1} \int_{1}^{3} \int_{-1}^{2}(2 x+3 y-z) d x d y d z$$

Find the absolute maximum and minimum values of cach function, subject to the given constraints. \(f(x, y)=x^{2}+y^{2}-2 x-2 y ; \quad x \geq 0, y \geq 0, x \leq 4\) and \(y \leq 3\)

Find the relative maximum and minimum values and the saddle points. $$\begin{aligned}S(b, m)=&(m+b-72)^{2}+(2 m+b-73)^{2} \\\&+(3 m+b-75)^{2}\end{aligned}$$

Labor force. The minimum hourly wage in the United States has grown over the years, as shown in the table below. $$\begin{array}{|r|c|}\hline \begin{array}{c}\text { Number of } \\\\\text { Years, } x,\end{array} & \begin{array}{c}\text { Minimum } \\\\\text { Hourly Wage }\end{array} \\\\\text { since } 1990 & \text { Saso } \\\0 & \$ 3.80 \\\1 & 4.25 \\\6 & 4.75 \\\7 & 5.15 \\\17 & 5.85 \\\18 & 6.55 \\\19 & 7.25 \\\\\hline\end{array}$$ a) For the data in the table, find the regression line, \(y=m x+b\) b) Use the regression line to predict the minimum hourly wage in 2015 and 2020
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

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.