Math Problem Statement
Solution
To analyze the given problem, let’s extract the details and solve for the domain of existence of each function.
Problem:
You need to determine and represent the domains of the following functions:
Solution:
1.
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Denominator: For the square roots to be defined, we require: This gives:
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Domain of : The function is defined in the open square:
2.
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Natural Logarithm: The logarithmic term requires its argument to be strictly positive:
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Parentheses: The factor must also be strictly positive:
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Combining conditions: The domain is defined by the intersection of these two regions:
3.
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Cotangent Function: The cotangent is undefined where its argument is zero:
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Domain of : This excludes all points where :
Final Result:
Let me know if you would like detailed graph representations or further clarifications.
Expanding Further:
- How can we graphically represent the domains for each function?
- What happens at boundary points of the domains?
- How do we analyze the interaction of and ?
- Why does exclude ?
- What are some numerical examples for valid and invalid points in each domain?
Tip: When working with multi-variable domains, break conditions into manageable parts and solve systematically!
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Math Problem Analysis
Mathematical Concepts
Domain of a function
Inequalities
Natural logarithm
Trigonometric functions
Formulas
sqrt(a): square root function
ln(a): natural logarithm
cot(a): cotangent function
Theorems
Square root existence requires non-negative arguments
Natural logarithm requires positive arguments
Cotangent is undefined at multiples of π
Suitable Grade Level
Grades 11-12
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