91Ó°ÊÓ

Rational functions are a key concept in mathematics and are defined as a fraction where both the numerator and the denominator are polynomials. For example, in the function given in the original exercise, \( f(x) = \frac{2x}{x-3} \), the numerator is \( 2x \) and the denominator is \( x-3 \). These functions are prevalent because they can model a variety of real-world scenarios, from physics to economics.
One of the important aspects of rational functions is understanding how their components behave. The numerator determines the output values or the range of the function, while the denominator affects the domain, particularly where the function can potentially be undefined. Evaluating where these functions are defined or undefined leads us to explore the concept of undefined values.

Undefined values occur in rational functions when the denominator equals zero. This is because division by zero is mathematically undefined, leading to a situation where the function can’t produce a real number output. In our specific example, \( f(x) = \frac{2x}{x-3} \), the denominator is \( x-3 \). If we set this equal to zero, which is \( x-3=0 \), we solve to find \( x=3 \).
At \( x=3 \), the function becomes \( \frac{2 \times 3}{3-3} \), which simplifies to \( \frac{6}{0} \), an undefined expression. Therefore, \( x=3 \) must be excluded from the function's domain. Identifying these values is crucial for understanding the full range of inputs over which the function operates correctly.

The concept of real numbers is foundational in defining the domain of a function. Real numbers include all the numbers on the number line, such as whole numbers, fractions, decimals, and irrational numbers. In mathematics, when we determine the domain of a function, we are looking for all possible values that a variable, such as \( x \), can take in the real number set.
In our function \( f(x) = \frac{2x}{x-3} \), the domain is expressed in terms of real numbers. While \( x=3 \) is where the function is undefined, the rest of the real numbers can be used as inputs for this function. Therefore, the domain can be expressed as the set of all real numbers except for 3: \( (-\infty, 3) \cup (3, \infty) \). This highlights how real numbers are used in defining the limits and possibilities of function inputs.