91Ó°ÊÓ

A rational function is a type of function that is expressed as the ratio of two polynomials. In simpler terms, it is a fraction where both the numerator and the denominator are polynomials. In the given function, \( g(x, y) = \frac{x-y}{x+y} \), we clearly see this structure. The numerator is \( x - y \) and the denominator is \( x + y \).

Rational functions are quite common in algebra and calculus because they help describe many natural phenomena. However, they can raise challenges because the denominator cannot be zero, which directly impacts the domain of the function.

The domain of a function is the set of all possible input values (in this case, \( x \) and \( y \)) for which the function is defined. Determining the domain is crucial because it lets us know where the function "works" without any problems.

For a two-variable function like \( g(x, y) = \frac{x-y}{x+y} \), we need to find all pairs \((x, y)\) such that the function produces a real number. This involves ensuring that the denominator is not zero, which brings us to the next important aspect.

In rational functions, denominator restriction is one of the most important considerations. This restriction means finding values of \( (x, y) \) that do not make the denominator zero.

For the function \( g(x, y) = \frac{x-y}{x+y} \), we have the denominator \( x + y \). To avoid division by zero (which is undefined), \( x + y \) must not equal zero. Therefore, we derive the line equation \( y = -x \).

Points on this line make the denominator zero and thus restrict the domain since the function is undefined at these points. Understanding this restriction is key to understanding where the function operates correctly.

Sketching a domain helps visualize where a function is defined, especially with two variables. In our case, for \( g(x, y) \), we begin by graphing the restriction.

Using the equation \( x + y = 0 \) or equivalently \( y = -x \), we sketch this line on a coordinate plane. Since \( x + y e 0 \) indicates the domain is all points except on this line, we draw \( y = -x \) as a dashed line. This visually denotes that points on it are excluded from the domain.

Finally, to complete the sketch, shade the regions on both sides of the dashed line. These shaded areas represent valid input pairs \( (x, y) \) that define the domain of the function, thus providing a clear visual guide to where the function is operational.