find and sketch the domain of the function f(x,y)= x/x^2+y^2

${\displaystyle f(x,y)=\frac{x}{x^2}+y^2}$

Cancel the common factor of $x$ and $x^2$.

Raise $x$ to the power of $1$.

${\displaystyle f(x,y)=\frac{x^1}{x^2}+y^2}$

Factor $x$ out of $x^1$.

${\displaystyle f(x,y)=\frac{x\cdot 1}{x^2}+y^2}$

Cancel the common factors.

${\displaystyle f(x,y)=\frac{1}{x^{}}+y^2}$

Subtract $y^2$ from both sides of the equation.

${\displaystyle f(x,y)-y^2=\frac{1}{x^{}}}$

Set the denominator in $\frac{1}{x^{}}$ equal to 0 to find where the expression is undefined.

$x=0$

The domain is all values of x that make the expression defined.

Interval Notation:

$(-\infty ,0)\cup (0,\infty )$

Set-Builder Notation:

$\{x|x\ne 0\}$