Question

how to find the domain and range of multivariable function

Answer 1

let f(x,y)=(x−1)(y−1)−−−−−−−−−−−√f(x,y)=(x−1)(y−1); what is the domain and the range?

I wanted to know if I was doing the domain part right and needed help with the range.

For the domain I got (x−1)(y−1)(x−1)(y−1) are good as long as they are any real number. is that right?



We need for f(x,y)=(x−1)(y−1)−−−−−−−−−−−√f(x,y)=(x−1)(y−1) to be defined. So we must have (x−1)(y−1)≥0(x−1)(y−1)≥0. Why?

Then recall: The domain of a function are those values on which the function is defined. So we know that f(x,y)f(x,y) is defined for all (x,y)(x,y) in the plane for which (x−1)(y−1)≥0.(x−1)(y−1)≥0. We can pick, say, any xx we want. This will then determine the values of yy on which the function f(x,y)f(x,y) is defined. We can determine yy as follows:

(x−1)<0⟹(y−1)≤0,(x−1)=0⟹y∈R,(x−1)>0⟹(y−1)≥0(x−1)<0⟹(y−1)≤0,(x−1)=0⟹y∈R,(x−1)>0⟹(y−1)≥0

All points (x,y)(x,y) satisfying the above inequalities are points in the domain of f(x,y)f(x,y), which will points in R2R2, the real plane. And it will follow that the range will necessarily be points in RR (the real line) such greater than or equal to zero: the set of all non-negative reals.
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