Math Problem Statement

Determine and represent the domains of existence of the functions: f(x, y) = 1 / (sqrt(1 - y^2) * sqrt(1 - x^2)), f(x, y) = ln(x^2 + 2y^2 - 1) * (8 - 2y - x^2 - y^2), f(x, y) = cot(xy).

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:

  1. f(x,y)=11y21x2f(x, y) = \frac{1}{\sqrt{1 - y^2} \sqrt{1 - x^2}}
  2. f(x,y)=ln(x2+2y21)(82yx2y2)f(x, y) = \ln(x^2 + 2y^2 - 1)(8 - 2y - x^2 - y^2)
  3. f(x,y)=cot(xy)f(x, y) = \cot(xy)

Solution:

1. f(x,y)=11y21x2f(x, y) = \frac{1}{\sqrt{1 - y^2} \sqrt{1 - x^2}}

  • Denominator: For the square roots to be defined, we require: 1y2>0and1x2>01 - y^2 > 0 \quad \text{and} \quad 1 - x^2 > 0 This gives: 1<y<1and1<x<1-1 < y < 1 \quad \text{and} \quad -1 < x < 1

  • Domain of f(x,y)f(x, y): The function is defined in the open square: D1={(x,y)R2:1<x<1,1<y<1}.D_1 = \{ (x, y) \in \mathbb{R}^2 : -1 < x < 1, -1 < y < 1 \}.


2. f(x,y)=ln(x2+2y21)(82yx2y2)f(x, y) = \ln(x^2 + 2y^2 - 1)(8 - 2y - x^2 - y^2)

  • Natural Logarithm: The logarithmic term requires its argument to be strictly positive: x2+2y21>0x2+2y2>1.x^2 + 2y^2 - 1 > 0 \quad \Rightarrow \quad x^2 + 2y^2 > 1.

  • Parentheses: The factor (82yx2y2)(8 - 2y - x^2 - y^2) must also be strictly positive: 82yx2y2>0.8 - 2y - x^2 - y^2 > 0.

  • Combining conditions: The domain is defined by the intersection of these two regions: D2={(x,y)R2:x2+2y2>1and82yx2y2>0}.D_2 = \{ (x, y) \in \mathbb{R}^2 : x^2 + 2y^2 > 1 \quad \text{and} \quad 8 - 2y - x^2 - y^2 > 0 \}.


3. f(x,y)=cot(xy)f(x, y) = \cot(xy)

  • Cotangent Function: The cotangent is undefined where its argument is zero: xynπfor integers n.xy \neq n\pi \quad \text{for integers } n.

  • Domain of f(x,y)f(x, y): This excludes all points where xy=nπxy = n\pi: D3={(x,y)R2:xynπ,nZ}.D_3 = \{ (x, y) \in \mathbb{R}^2 : xy \neq n\pi, \, n \in \mathbb{Z} \}.


Final Result:

  1. D1={(x,y):1<x<1,1<y<1}D_1 = \{ (x, y) : -1 < x < 1, -1 < y < 1 \}
  2. D2={(x,y):x2+2y2>1and82yx2y2>0}D_2 = \{ (x, y) : x^2 + 2y^2 > 1 \quad \text{and} \quad 8 - 2y - x^2 - y^2 > 0 \}
  3. D3={(x,y):xynπ,nZ}D_3 = \{ (x, y) : xy \neq n\pi, n \in \mathbb{Z} \}

Let me know if you would like detailed graph representations or further clarifications.


Expanding Further:

  1. How can we graphically represent the domains for each function?
  2. What happens at boundary points of the domains?
  3. How do we analyze the interaction of x2+2y2>1x^2 + 2y^2 > 1 and 82yx2y2>08 - 2y - x^2 - y^2 > 0?
  4. Why does cot(xy)\cot(xy) exclude xy=nπxy = n\pi?
  5. 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