91Ó°ÊÓ

A rational function is a type of function represented by the quotient of two polynomials. For instance, in our exercise, the function is:

\(f(x) = \frac{x}{(x-2)(x+3)}\).

Here, \(x\) is in the numerator and \((x-2)(x+3)\) is in the denominator. Understanding rational functions is essential as they often have restrictions on their domain due to division by zero.

To put it simply, rational functions can be thought of as fractions involving polynomials. The denominator of the fraction cannot be zero because division by zero is undefined. This brings us to the next core concept:

In mathematics, we often encounter fractions. A fraction becomes undefined when its denominator equals zero. For example, in the function given:

\(f(x) = \frac{x}{(x-2)(x+3)}\)

The denominator is \((x-2)(x+3)\). To find out when the function becomes undefined, we need to determine when the denominator is zero:

\( (x-2)(x+3) = 0 \)

This shows us which values of \(x\) make the denominator zero. Solving \((x-2) = 0\) gives \(x = 2\), and solving \((x+3) = 0\) gives \(x = -3\). These are the values where the function becomes undefined, as they make the denominator zero. It's crucial to note these values because they lead directly to our final core concept:

The domain of a function consists of all possible input values (\(x\)) that will output real numbers. For rational functions, the domain is restricted by values that make the denominator zero. From our previous discussion, we know the function:

\(f(x) = \frac{x}{(x-2)(x+3)}\)

is undefined at \(x = 2\) and \(x = -3\). Therefore, these values must be excluded from the domain.

When writing the domain, we say:

\( x \in \mathbb{R} \) except \( x = 2 \) and \( x = -3 \).

This notation signifies that \(x\) can be any real number except 2 and -3. Understanding domain restrictions helps in graphing functions and solving equations involving rational functions, making it an essential concept in algebra and calculus.