What Are the Domain and Range of the Function f(x,y) = ln(y-2x)?

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SUMMARY

The function f(x,y) = ln(y-2x) has a domain defined by the condition y - 2x > 0, leading to the largest possible domain being y > 2x. The range of the function, when defined over this domain, is (-∞, ∞). Additionally, for f(x,y) to be greater than zero, the condition y - 2x must be greater than 1, resulting in the constraint y > 2x + 1. The discussion also clarifies that two different level curves cannot intersect, as this would violate the definition of a function, which mandates a unique output for each input.

PREREQUISITES
  • Understanding of logarithmic functions, specifically ln(x)
  • Familiarity with the concept of domain and range in multivariable functions
  • Knowledge of level curves and their significance in function analysis
  • Basic algebraic manipulation skills to solve inequalities
NEXT STEPS
  • Study the properties of logarithmic functions, focusing on their domains and ranges
  • Learn about multivariable calculus, particularly the analysis of functions of two variables
  • Explore the concept of level curves in detail and their implications in graphical representations
  • Investigate inequalities involving logarithmic functions and their applications in real-world scenarios
USEFUL FOR

Students and educators in mathematics, particularly those studying calculus and functions of multiple variables, as well as anyone interested in understanding the behavior of logarithmic functions in a multivariable context.

Saunderssim
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Hi,

Kinda need help for this question.

f(x,y) = ln(y-2x)

1. Find the largest possible domain
2. Find the range of the function where the function is defined over the largest possible domain
3. find the largest possible domain if it is desired that f(x,y) > 0

Explain why two different level curves cannot intersect.
 
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Saunderssim said:
Hi,

Kinda need help for this question.

f(x,y) = ln(y-2x)

1. Find the largest possible domain
What is the largest possible domain of ln(x)?

2. Find the range of the function where the function is defined over the largest possible domain
What is the range of ln(x)?

3. find the largest possible domain if it is desired that f(x,y) > 0
For what values of x is ln(x)> 0?

Explain why two different level curves cannot intersect.
What is the definition of "level curves"? What would be true about a point at which two different level curves intersect? How does that contradict the definition of "function"?
 
domain: x>0
Range: (-infinity, infinity)
value: x>1

I don't quite know how to answer the last part of the question. Well a function mostly has one specific answer for every specific value...
 

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