Domain and range of functions of 2 variables

[rolls]

I'm doing some exam revision and there's a few questions on finding the domain and range of functions of 2 variables.

With a function of 1 variable I know that the domain and range is just finding the max and min x value, and the max and min y value.

With a function of 2 variables I am confused however, what exactly does it mean to find the domain and range, and what is the method for doing so. Eg this question.

h(x,y) = 1500 - 3x^2-5y^2

Domain = all real numbers in x,y
Range = (-oo,1500)

However I do not know the process for finding this out, mainly as I am confused to the definition of what range and domain actually is.

Can someone help me out here? More with what domain and range mean, I'm sure the question is easy once I know this.

 

[HallsofIvy]

I'm doing some exam revision and there's a few questions on finding the domain and range of functions of 2 variables.

With a function of 1 variable I know that the domain and range is just finding the max and min x value, and the max and min y value.

No, you don't know that- or shouldn't. The domain of a function is either given explicitely or is assumed to be all x-values for which the formula defining the function can be calculated. For example, f(x)=2+ 1/(x-1) can be calculated for all x except 1- we can't divide by 0- so the domain is "all real numbers except 1". That has nothing to do with "max and min x values". The range of a function is the set of all possible y values for those x values. Again, that has nothing to do with "max and min y values". In the case above, y= f(x)= 2+ 1/(x-1), y has no max or min value- it can be arbitrarily high or arbitrarily low- but it cannot be 2 because a fraction is 0 only if the numerator is 0. The numerator of 1/(x-1) is never 0 so 1/(x-1) is never 0 and 2+ 1/(x-1) is never 2. The range is "all real numbers except 2.

With a function of 2 variables I am confused however, what exactly does it mean to find the domain and range, and what is the method for doing so. Eg this question.

h(x,y) = 1500 - 3x^2-5y^2

Domain = all real numbers in x,y
Range = (-oo,1500)

However I do not know the process for finding this out, mainly as I am confused to the definition of what range and domain actually is.

Can someone help me out here? More with what domain and range mean, I'm sure the question is easy once I know this.

And I feel sure your textbook has the definitions in it! In this particular case, we can square any number, multiply any number by 3 or 5, and subtract any number from 1500. The domain is all of R x R or all pairs of real numbers (x, y).

Because a square is never negative, -3x²- 5y² is never positive so we can never get a value larger than 1500. The range is all real numbers equal to or less than 1500. Using standard notation, the range is NOT (-oo, 1500) as you have, it is (-oo, 1500], "]" meaning that the endpoint, 1500 is included in the set.

 

[rolls]

Ah fantastic, thank you for the informative reply. So the domain is the range of values that give you a real answer eg 0/0 is not an answer, and the range is the range of all possible outputs.

I would check my textbook, but I do not own it. Thank you :)