Calculus Discussion 15

Question 7

\[ f(x,y) = x^2 + y^2 + 2 \]

  • Domain: All real numbers
  • Range: [2, inf)

Question 8

\[ f(x,y) = x + 2y \]

  • Domain: All real numbers
  • Range: All real numbers

Question 9

\[ f(x,y) = x - 2y \]

  • Domain: All real numbers
  • Range: All real numbers

Question 10

\[ f(x,y) = \frac{1}{x+2y} \]

  • Domain of x: x is in (-inf, -x/2) or (-x/2, inf)
  • Domain of y: y is in (-inf, -x/2) or (-x/2, inf)
  • Range: All real numbers

Question 11

\[ f(x,y) = \frac{1}{x^2+y^2+1} \]

  • Domain: All real numbers
  • Range: [1, inf)

Question 12

\[ f(x,y) = sinx cosy \]

  • Domain: All real numbers
  • Range: [-1, 1]

Question 13

\[ f(x,y) = \sqrt{9 - x^2 - y^2} \]

  • Domain of x: x is in [-3 + |y|,3 - |y|]
  • Domain of y: y is in [-3 + |x|,3 - |x|]
  • Range: [0, 3]

Question 14

\[ f(x,y) = \frac{1}{\sqrt{x^2 + y^2 - 9}} \]

  • Domain of x: x is in (-inf,-3 + |y|) or (3 - |y|, inf)
  • Domain of y: y is in (-inf,-3 + |x|) or (3 - |x|, inf)
  • Range: (0, inf)