1. ## Domain and Range

Hi,

I'm just wondering if I'm correct and also what notation I should use here.

The question is asking for the domain and range of $\displaystyle f(x,y)=e^{y-x^2}$

I think the domain is $\displaystyle Domain \subset R^2$ and $\displaystyle Range>0$

2. ## Re: Domain and Range

yes, the domain is a subset of $\displaystyle \mathbb{R}^2$. but in this case, if f(x,y) makes sense for all real numbers x and y,

then the domain equals $\displaystyle \mathbb{R}^2$.

now $\displaystyle e^x$ is always positive, so we know the range is a subset of the positive reals. the question is, can $\displaystyle y-x^2$

turn out to be any real number? and it can: let's say we have a certain real number r in mind. then we could pick x = 0, and y = r.

so now we ask, given that $\displaystyle y-x^2$ can be any real number, can $\displaystyle e^{y-x^2}$ be any positive real number?

suppose s is positive. then ln(s) makes sense, so we can pick (x,y) = (0,ln(s)), and then $\displaystyle e^{\ln(s)-0} = e^{\ln(s)} = s$.

so we see any positive real number is in the range of f, and only positive real numbers are in the range of f.

so range(f) = $\displaystyle \{x \in \mathbb{R}: x > 0\}$. this is sometimes also written: $\displaystyle \mathbb{R}^+$ or as the interval $\displaystyle (0,\infty)$

3. ## Re: Domain and Range

Not sure what your answer is for the Domain: it's all of $\displaystyle \mathbb{R}^2$.