Question

Find and sketch the domain of the function.\(f(x,y) = \frac{{\sqrt {y - {x^2}} }}{{(1 - {x^2})}}\).

Short answer

The domain of the function \(f(x,y)\)is \(\{ (x,y)/y \ge {x^2},x \ne \pm 1\} \).

Step-by-step solution

Step 1:- Consideration

Let \(f(x,y) = \frac{{\sqrt {y - {x^2}} }}{{(1 - {x^2})}}\)is a function.

Step 2:- Finding the domain

The function\(f(x,y)\)is real if

\(\sqrt {y - {x^2} > 0} \)and\(1 - {x^2} \ne 0\)

\(\sqrt {y - {x^2} > 0} \)if\(y - {x^2} \ge 0\)

\(\begin{aligned}{l} &\Rightarrow y \ge {x^2}\\(1 - {x^2}) &\ne 0\\ \Rightarrow {x^2} &\ne 1\\ \Rightarrow x &\ne \pm 1\end{aligned}\)

The lines \(x \ne \pm 1\)are excluded from the domain from \(y \ge {x^2}.\)

Step 3:- Graph

The domain of the function can be find through the graph also

Hence, the domain of the function \(f(x,y) = \frac{{\sqrt {y - {x^2}} }}{{(1 - {x^2})}}\) is \(\{ (x,y)/y \ge {x^2},x \ne \pm 1\} \)