91Ó°ÊÓ

In mathematics, a **function of two variables** is a rule that assigns a unique output to each pair of inputs. For example, the given function is: \( g(x, y) = \frac{1}{y + x^2} \).
Here, the output depends on the two inputs, \( x \) and \( y \). This means that different pairs of \( (x, y) \) will result in different outputs. Such functions are commonly used to describe surfaces or other multidimensional relationships. To visualize this, imagine a 3D surface where \( x \) and \( y \) are coordinates on the plane, and the function value \( g(x, y) \) is the height of the surface at those coordinates.
When dealing with these functions, it is crucial to understand how both variables impact the output. It is also important to look at any restrictions on these variables.

One important aspect to identify for functions involving fractions is the **denominator restrictions**. A function is undefined if its denominator equals zero because division by zero is mathematically undefined. For the given function \( g(x, y) = \frac{1}{y + x^2} \), we need to ensure that the denominator \( y + x^2 \) is never zero.
To find when the denominator is zero, set the denominator equal to zero and solve for \( y \): \[ y + x^2 = 0 \] Solving this, we get: \[ y = -x^2 \] This equation tells us the values of \( x \) and \( y \) that will make the denominator zero, causing the function to be undefined. In simpler terms, any pair \( (x, y) \) where \( y = -x^2 \) will break our function.

The domain of a function consists of all possible input values (here, pairs \( (x, y) \)) that result in a real and finite output. To determine the **domain** of the function \( g(x, y) = \frac{1}{y + x^2} \), we need to analyze the denominator restrictions identified earlier.
From the previous section, we know that the function is undefined when \( y = -x^2 \). Therefore, we must exclude these (x, y) pairs from the domain. The domain of \( g(x, y) \) will be all real numbers except when \[ y = -x^2 \]
So, we can express the domain of the function as follows: \[ \text{Domain: } \ \text{All pairs } (x, y) \text{ where } y eq -x^2 \] By understanding and applying these steps, you can easily solve for the domain of similar functions.