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Question #18
Hi Randy,

I actually don't get domains for multivariable calculus and level curves.Sorry, I am actually stuck on question #18 attached below and need some help on it.

Many Many Thanks,
Sam

For #18, you have to consider what types of inputs are not allowed.

Because there is a fraction, you cannot divide by zero, which means x-y cannot be zero.

That means x=y is forbidden. The domain is the set of all pairs (x,y) that are not equal.

The range can is all reals, because if you want some particular z value to be in the range:

z = (x+y)/(x-y)

you can just solve for x:

x = y (z+1)/(z-1)

This works for all possible z, unless z=1, but you can easily get z=1 too, with x=1, y=0.

For #16, you can put anything into an exponent so the domain is all pairs (R⊃).

The range is a bit tricky. Exponentials are always positive, so the range can only have z>0. But also, x^2 + y^2 is always positive, so the exponential is always negative (or zero). That means z ≤ 1. So the range is the interval (0,1].

For #14, you need what's in the square root to be non-negative, so:

9 - x² - y² ≥ 0

which is really interior of a circle with radius 3:

x² + y² ≤ 9

The range is [0,3], because what's in the square root cannot exceed 9 (at x=0,y=0) and can't get less than 0 without becoming imaginary.

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