# Domain and Range of f(x,y)

• Oct 1st 2008, 08:31 PM
JonathanEyoon
Domain and Range of f(x,y)
I'm having trouble understanding how to get these from these types of problems.

What should I be looking for and how. Any hints, tips , or methods?

Some examples of course would be

1. e^sqrt(z - x^2 - y^2)

2. ln(25 - x^2 - y^2 - z^2)

3. sqrt( 1 + x - y^2)

More than just answering these problems i'm more interested in knowing how to derive the domains and ranges since i'm really gonna need to know it. Thanks in advance guys (Worried)
• Oct 1st 2008, 08:50 PM
icemanfan
1. Identify that the square root is nonnegative and that the expression under the square root must be nonnegative. So the domain is all real x, y, z such that $\displaystyle z - x^2 - y^2 \geq 0$, and the range is $\displaystyle f(x,y,z) \geq 1$, since the smallest value for the exponent is 0, and there is no largest value for the exponent.

2. The natural logarithm only takes positive arguments. Hence the domain is all real x, y, z such that $\displaystyle x^2 + y^2 + z^2 < 25$. The range is $\displaystyle 0 < f(x,y,z) \leq \ln 25$.

3. Again, the square root is nonnegative and the expression under the square root must be nonnegative. So the domain is all real x, y such that $\displaystyle 1 + x - y^2 \geq 0$, and the range is $\displaystyle 0 \leq f(x, y)$, since $\displaystyle 1 + x - y^2$ is unbounded from above.
• Oct 1st 2008, 09:00 PM
JonathanEyoon
mMmm... ok

Uhmmm I see from what you said it's pretty easy but say I didn't give you any of those problems. Can you explain what i'm supposed to look for, how I should be interpreting the functions given and so on?
• Oct 1st 2008, 09:11 PM
icemanfan
Basically, you need to recognize when a function would restrict domain. Some functions restrict domain and some don't. If you have a square root function or a logarithm function, it will have a restricted domain, as you have seen. Also, if a function is of the form $\displaystyle \frac{f(x,y,z)}{g(x,y,z)}$, then $\displaystyle g(x,y,z)$ is not allowed to be zero.
The domain of a polynomial function is all real numbers.

As for determining the range: recognize that any terms that are squared (or raised to any even power) must be positive numbers; also recognize that exponential functions must produce positive numbers. There are several more rules for functions in general but basically you need to find absolute minima and absolute maxima if they exist, and for a lot of multivariable functions that requires multivariable calculus.
• Oct 1st 2008, 09:20 PM
JonathanEyoon
Thanks alot~ I got it (Rock)