Question

How do you find the domain of f(r,s)=sqrt(1-r)-e^(r/s)?

Answer 1

\[f(r,s)=\sqrt{1-r}-e^{r/s}\]
correct?

Answer 2

yes

Answer 3

Okay, what's the domain of the square root function?
Given \(f(x)=\sqrt x\), clearly, the domain is \(x\ge0\), right? How would this translate to your function?

Answer 4

For the first term, that is.

Answer 5

r cannot be greater than one?

Answer 6

Right!

Now onto the next term. What do you know about the exponential function, \(f(x)=e^x\) ? What's the domain here, in terms of \(x\)?

Answer 7

all real numbers?

Answer 8

Right again! However, the exponent isn't just \(x\), or \(r\) or \(s\).

Since it's \(\dfrac{r}{s}\), you have to correct for this. What values of \(r\) or \(s\) can't be used here?

Answer 9

well, r can be anything...even zero. but s can't be zero?

Answer 10

Yes. So what would the overall domain of \(f(r,s)\) be?

Answer 11

\[s \neq 0\]and\[r \ge 1\]but is there a certain way that it needs to be formatted to be "correct?"

Answer 12

Correct! There are a few ways of writing the domain set, but here's one of the more common forms for 2-variable functions like this one:
\[f(x)=\left\{(r,s)\in\mathbb{R}^2~|~r\ge1,~s\not=0\right\}\]
Which would translate, roughly, to
"the set of all coordinate points in the x-y (r-s, or real) plane such that r is at least 1 and s is non-zero."

Answer 13

okay, thank you so much!

Answer 14

You're welcome!