Q 47. - Calculus

Question

Determine the domains of the functions in Exercises 47–56, and find where the functions are continuous. 

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

Step-by-Step Solution

Answer

The function $f(x,y)=\frac{x^2}{x^2-y^2}$ is continuous on the set $\left\{\left(x,y\right)\in f^2:x^2\ne y^2\right\}$.

Step 1. Given information.

We have given expression: $f(x,y)=\frac{x^2}{x^2-y^2}$

Step 2: Determine the domains of the functions.

Consider the function:$g:f^2\to f$. Then the domain of the function is

$Domain \left(g\right)=\left\{\left(x,y\right)\in f^2:g(x,y) is defined\right\}$

Since the rational function $f(x,y)=\frac{x^2}{x^2-y^2}$ is defined for all $\left(x,y\right)\in f^2$ such that

$$\begin{gathered} x^2-y^2\ne 0 \\ {x}^2\ne y^2 \end{gathered}$$

The domain of the function is  $Domain\left(f\right)=\left\{\left(x,y\right)\in f^2:x^2\ne y^2\right\}$
.

Step 3. To find continuous of the function.

Since $x^2$ and $x^2-y^2$ being a polynomial function of two variable is continuous for every point on $f^2$.

The rational function is continuous where all those points where $\frac{x^2}{x^2-y^2}$ is defined.

The rational function is discontinuous only at the points where$x^2-y^2=0$ that is $x=y$.

Hence the given function $f(x,y)=\frac{x^2}{x^2-y^2}$ is continuous on the set$\left\{\left(x,y\right)\in f^2:x^2\ne y^2\right\}$.