91Ó°ÊÓ

In mathematics, a function is said to be undefined for certain inputs when the result does not produce a valid numerical output. Understanding where a function is undefined is crucial because these points determine the limits of the function's domain. For the function given in our exercise, \( f(x, y)=\frac{3xy}{y-x^{2}} \), it becomes undefined when the denominator \( y-x^{2} \) equals zero, as dividing by zero is not allowed in mathematics.
This leads us to set the expression \( y-x^{2} \) equal to zero. Solving this equation gives \( y = x^{2} \). Therefore, any point \((x, y)\) where \( y = x^{2} \) is where the function is undefined. These points form a boundary in the xy-plane that must be taken into account when determining where the function can operate.

The concept of real values is important when discussing the domain of a function. Real numbers include all the numbers on the number line, encompassing both negative and positive numbers, as well as all the fractions and decimals in between. In the context of our function, we must identify where both \( x \) and \( y \) are real numbers, ensuring we consider the full range of possible input values.
The domain consists of all possible real values for \( x \) and \( y \) except where the undefined point \( y = x^{2} \) exists. Thus, for every real \( x \), there is a range of real \( y \) values that are valid, except where \( y \) exactly equals \( x^{2} \).

• All real values of \( x \) are considered.
• All real values of \( y \) are considered, except \( y eq x^{2} \).

These conditions ensure the function remains defined and operational, providing a clear understanding of its domain in the real number system.

A parabola is a specific type of curve on a graph, and it has significant importance when discussing the domain of our function. The equation \( y = x^{2} \) represents a parabola that opens upwards with its vertex at the origin \((0, 0)\). Parabolas are symmetric around their vertical axis, in this case, the y-axis.The parabola \( y = x^{2} \) is the delineating line separating the valid and invalid regions for our function. Points lying directly on this curve render the function undefined. Thus, understanding the shape and location of this parabola helps in visualizing which parts of the plane the function can operate in.
It separates the plane into two regions:

• Points above the parabola \( y > x^{2} \).
• Points below the parabola \( y < x^{2} \).

In the exercise, the parabola acts as the boundary that is excluded from the domain, making it pivotal in the function’s graphical representation.

Graphically representing the domain of a function can provide a clearer, visual understanding of where the function is defined. For our function, visualizing the domain involves plotting the parabola \( y = x^{2} \), but realizing that this curve itself is excluded from the domain.
Sketch the curve without fully drawing it, indicating it as a dashed line perhaps. Highlighting the area above and below the parabola demonstrates the valid regions for \( (x, y) \). These regions encompass all points that do not lie directly on the parabola. By doing so,

• The space above \( y = x^{2} \) reflects valid points \((x, y)\) where \( y > x^{2} \).
• The space below reflects valid points where \( y < x^{2} \).

Visual tools like graphs are instrumental in grasping complex algebraic concepts, providing students with an intuitive view of mathematical domains.