# Thread: Domain of a function

1. ## Domain of a function

f(x,y) = Sqrt(y-x^2)/(x^2+(y-1)^2) Find and graph the domain of the function

f(x,y) = Sqrt(y-x^2)/(x^2+(y-1)^2) Find and graph the domain of the function
The domain of $\displaystyle \sqrt{x}$ is all $\displaystyle x\ge 0$ so you must have $\displaystyle y- x^2\ge 0$. Graph $\displaystyle y= x^2$. $\displaystyle y\ge x^2$ for all (x,y) below that graph. The domain of 1/x is all x not equal to 0. The domain of $\displaystyle \frac{1}{x^2+(y-2)^2}$ is all (x, y) except the single point where that denominator is 0. The domain of f is the points that are in both of those domains.

Find and graph the domain of: .$\displaystyle f(x,y) \;=\; \sqrt{\frac{y-x^2}{x^2+(y-1)^2}}$

The denominator is always positive except at (0,1) where the function is undefined.

The numerator must be nonnegative: .$\displaystyle y - x^2 \:\geq \:0 \quad\Rightarrow\quad y \:\geq \:x^2$
. . This is the set of points on and above the parabola: $\displaystyle y \,=\,x^2$

The region looks like this:
Code:
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*::::::::o::::::::*
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*:::|:::*
- - - - - - * - - - - - -
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where there is a "hole" at (0,1).