Question

How do you find the domain and range of `f(x,y) = (x-3)^2 /4 - (y+1)^2 /16`?

Answer 1

Please consider the x term. Is there any real number that one can substitute into that term which would make the function be undefined. No. Therefore, x is any real number:

`x in RR`

The y term is similar to the x term, except that is has a negative sign, therefore, the same is true for the y term:

`x,y in RR`

This constitutes the domain.

For the range, please consider:

`f(3,-1) = (3-3)^2 /4 - ((-1)+1)^2 /16`

`f(3,-1)= 0`

Can we do something that makes the function start from zero and approach positive infinity? Yes. We can hold y constant at -1 and let `x to +-oo`. Therefore, the range is at least all positive real numbers.

Can we do something that makes the function start from zero and approach negative infinity? Yes. We can hold x constant at 3 and let `y to +-oo`. This adds the negative real numbers to the range:

`f(x,y) in RR`