91Ó°ÊÓ

The denominator of a function plays a crucial role in defining its domain because a division by zero is undefined in mathematics. For the function \(f(x, y) = \frac{\sqrt{y} - x^2}{1 - x^2}\), we first need to ensure that the denominator \(1 - x^2\) never equals zero. This requirement leads us to solve the equation \(1 - x^2 = 0\). By rearranging, we get \(x^2 = 1\), which implies two critical values: \(x = 1\) and \(x = -1\). These values cause the denominator to be zero, making the function undefined at these points.

When identifying restrictions due to the denominator, always remember:

• If \(D(x) = 0\), then \(x\) is a restriction.
• Solve \(D(x) = 0\) to find restricted values.
• Exclusion of these values from the domain is necessary for the function to remain valid.

In this function, values \(x = 1\) and \(x = -1\) are not included in the domain. Hence, they are our denominator restrictions.

Square root expressions introduce a different kind of restriction, mainly because the square root of a negative number isn't defined within the set of real numbers. In the function \(f(x, y) = \frac{\sqrt{y} - x^2}{1 - x^2}\), the numerator contains \(\sqrt{y}\), which necessitates that \(y \geq 0\). This is because the domain of the square root function is the set of non-negative numbers.

When dealing with square roots in functions, it's vital to:

• Set up inequalities to ensure non-negativity. For \(\sqrt{y}\), ensure \(y \geq 0\).
• Confirm that any terms moved outside the square root don't add negative values back.
• Combine these constraints with any other existing ones to find the valid domain.

By identifying such constraints, the domain is refined further, ensuring only valid inputs are used.

Sketching a graph of the domain is a useful visual aid. It helps in understanding the feasible values of \(x\) and \(y\) for continuous functions. For our function, \(f(x, y) = \frac{\sqrt{y} - x^2}{1 - x^2}\), combining the restrictions from the denominator and the square root, the domain becomes all points \((x, y)\) such that \(y \geq 0\), \(x eq 1\), and \(x eq -1\).

To sketch:

• Start by drawing the y-axis where \(y \geq 0\) since negative y-values aren't allowed due to the square root.
• Exclude the vertical lines where \(x = 1\) and \(x = -1\).
• The remaining shaded area on the plane represents the domain satisfying all conditions.

By visualizing, we clearly see where the function is defined and recognize any gaps or restrictions, such as the lines that are omitted. This can aid greatly in further analysis or function manipulation.