Question

For a function g(r,s)=arccos(rs), describe the domain and range.

Answer 1

\[\frac{ -1 }{ s } \le r \le \frac{ 1 }{ s }\]

Answer 2

Is that a valid method of describing the domain?

Answer 3

the graph of y=cos(x) is restricted to the domain [0,pi] with range [-1,1]
such that y=arccos(x) can exist and y=arccos(x) has domain [-1,1] and range [0,pi]
so you have
\[-1 \le rs \le 1 \]

Answer 4

your inequality assumes s is positive
what would you do if s was negative?

Answer 5

You're allowed to do that. restrict r and s in the same equation. Hmm, I didn't know. I got my answer from your answer, but I thought I would have to isolate a variable for some reason.

Answer 6

I don't know what your orders are but I would simply say the domain is \[rs \in [-1,1]\]

you could isolate a variable you would just have to do cases

Answer 7

Okay thanks.

Answer 8

\[\text{ if } s>0 \text{ then } \frac{-1}{s} \le r \le \frac{1}{s} \\ \text{ if } s<0 \text{ then } \frac{-1}{s} \ge r \ge \frac{1}{s}\]

Answer 9

and I guess we would have to do a separate case for s=0 since you cannot divide by 0

Answer 10

\[\text{ if } s=0 \text{ then it doesn't matter what } r \text{ is } \]

Answer 11

Because rs would be 0?

Answer 12

right and arccos(0) exist so s=0 where r=anything is in the domain

Answer 13

Oh got it. Thanks.

Answer 14

but for the r=any real number we must have s=0 at the time

Answer 15

so again we could write this:
\[\text{ if } s>0 \text{ then } \frac{-1}{s} \le r \le \frac{1}{s} \\ \text{ if } s<0 \text{ then } \frac{-1}{s} \ge r \ge \frac{1}{s} \\ \text{ if } s=0 \text{ then } r \in R\]

Answer 16

I bet you remember what the graph of f(x)=1/x looks like

Answer 17

and the graph of f(x)=-1/x

Answer 18

you would want r to be between all of these thingys if that makes sense