Math Problem Statement

Is the function f(x, y) = 1 / sqrt(-y^2 - 5 * x^2 + 166) defined at the given points?

Solution

The task here involves determining whether the function f(x,y)=1y25x2+166f(x, y) = \frac{1}{\sqrt{-y^2 - 5 \cdot x^2 + 166}} is defined at specific points given in the options.

For the function f(x,y)f(x, y) to be defined at a point (x,y)(x, y), the expression inside the square root, y25x2+166-y^2 - 5 \cdot x^2 + 166, must be non-negative. Specifically, we need:

y25x2+1660-y^2 - 5 \cdot x^2 + 166 \geq 0

or equivalently,

5x2+y2166.5x^2 + y^2 \leq 166.

Now, we will check each point given in the options to see if it satisfies this inequality.

Checking Each Option

  1. Option a) (2,13)(2, -13): 5(2)2+(13)2=54+169=20+169=189(not valid)5 \cdot (2)^2 + (-13)^2 = 5 \cdot 4 + 169 = 20 + 169 = 189 \quad (\text{not valid})

  2. Option b) (3,11)(-3, -11): 5(3)2+(11)2=59+121=45+121=166(valid)5 \cdot (-3)^2 + (-11)^2 = 5 \cdot 9 + 121 = 45 + 121 = 166 \quad (\text{valid})

  3. Option c) (5,3)(5, 3): 5(5)2+(3)2=525+9=125+9=134(valid)5 \cdot (5)^2 + (3)^2 = 5 \cdot 25 + 9 = 125 + 9 = 134 \quad (\text{valid})

  4. Option d) (4,9)(-4, -9): 5(4)2+(9)2=516+81=80+81=161(valid)5 \cdot (-4)^2 + (-9)^2 = 5 \cdot 16 + 81 = 80 + 81 = 161 \quad (\text{valid})

  5. Option e) (5,6)(-5, -6): 5(5)2+(6)2=525+36=125+36=161(valid)5 \cdot (-5)^2 + (-6)^2 = 5 \cdot 25 + 36 = 125 + 36 = 161 \quad (\text{valid})

Conclusion

The function f(x,y)f(x, y) is defined at points b), c), d), and e). Only option a) does not satisfy the inequality. Therefore, the correct answers would be b, c, d, and e.

Would you like further details or have any specific questions?

Here are some related questions to expand on this topic:

  1. How can inequalities involving multiple variables be visualized on a graph?
  2. What happens to the function f(x,y)f(x, y) if the inequality becomes an equality?
  3. How do different values in the function affect the domain constraints?
  4. Can similar methods be used to determine the domain of more complex functions?
  5. What are some practical applications of defining functions with restricted domains?

Tip: When analyzing inequalities involving functions, always simplify the expression to make the conditions for the function's domain more apparent.

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Inequalities
Function Domains

Formulas

f(x, y) = 1 / sqrt(-y^2 - 5 * x^2 + 166)
Inequality condition: -y^2 - 5 * x^2 + 166 >= 0

Theorems

Domain of a function with a square root in the denominator

Suitable Grade Level

Grades 11-12