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Find the domain and range and describe the level curves for the function f(x,y). f(x.y) = In (6x + 7y) O A. Domain: all points in the xy-plane satisfying 6x + 7y > 0 Range: all real numbers Level curves: lines 6x + 7y = c O B. Domain: all points in the xy-plane satisfying 6x + 7y > 0 Range: real numbers z20 Level curves: lines 6x + 7y = c O C. Domain: all points in the xy-plane Range: all real numbers Level curves: lines 6x + 7y = c O D. Domain: all points in the xy-plane satisfying 6x+ Range: all real numbers Level curves: lines 6x + 7y =c x+7y 20

Find the domain and range and describe the level curves for the function f(x,y). f(x.y) = In (6x + 7y) O A. Domain: all points in the xy-plane satisfying 6x + 7y > 0 Range: all real numbers Level curves: lines 6x + 7y = c O B. Domain: all points in the xy-plane satisfying 6x + 7y > 0 Range: real numbers z20 Level curves: lines 6x + 7y = c O C. Domain: all points in the xy-plane Range: all real numbers Level curves: lines 6x + 7y = c O D. Domain: all points in the xy-plane satisfying 6x+ Range: all real numbers Level curves: lines 6x + 7y =c x+7y 20

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Title: Analyzing Domain and Range with Level Curves for Multivariable Functions --- **Problem Statement:** Determine the domain and range and describe the level curves for the function \( f(x, y) = \ln (6x + 7y) \). **Options:** - **A.** - **Domain:** All points in the xy-plane satisfying \(6x + 7y > 0\) - **Range:** All real numbers - **Level Curves:** Lines \(6x + 7y = c\) - **B.** - **Domain:** All points in the xy-plane satisfying \(6x + 7y \geq 0\) - **Range:** Real numbers \(z \geq 0\) - **Level Curves:** Lines \(6x + 7y = c\) - **C.** - **Domain:** All points in the xy-plane - **Range:** All real numbers - **Level Curves:** Lines \(6x + 7y = c\) - **D.** - **Domain:** All points in the xy-plane satisfying \(6x + 7y \geq 0\) - **Range:** All real numbers - **Level Curves:** Lines \(6x + 7y = c\) --- **Explanation of Diagrams:** The function \( f(x, y) = \ln (6x + 7y) \) involves level curves represented by lines \(6x + 7y = c\). The domain requires \(6x + 7y\) to be greater than zero because the logarithm function is undefined for non-positive values. Therefore, the level curves are straight lines, and their specific properties depend on the given constraints in each option. This exercise explores the relationships between domain, range, and level curves in two-variable logarithmic functions.
Transcribed Image Text:Title: Analyzing Domain and Range with Level Curves for Multivariable Functions --- **Problem Statement:** Determine the domain and range and describe the level curves for the function \( f(x, y) = \ln (6x + 7y) \). **Options:** - **A.** - **Domain:** All points in the xy-plane satisfying \(6x + 7y > 0\) - **Range:** All real numbers - **Level Curves:** Lines \(6x + 7y = c\) - **B.** - **Domain:** All points in the xy-plane satisfying \(6x + 7y \geq 0\) - **Range:** Real numbers \(z \geq 0\) - **Level Curves:** Lines \(6x + 7y = c\) - **C.** - **Domain:** All points in the xy-plane - **Range:** All real numbers - **Level Curves:** Lines \(6x + 7y = c\) - **D.** - **Domain:** All points in the xy-plane satisfying \(6x + 7y \geq 0\) - **Range:** All real numbers - **Level Curves:** Lines \(6x + 7y = c\) --- **Explanation of Diagrams:** The function \( f(x, y) = \ln (6x + 7y) \) involves level curves represented by lines \(6x + 7y = c\). The domain requires \(6x + 7y\) to be greater than zero because the logarithm function is undefined for non-positive values. Therefore, the level curves are straight lines, and their specific properties depend on the given constraints in each option. This exercise explores the relationships between domain, range, and level curves in two-variable logarithmic functions.
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