91Ó°ÊÓ

Rational functions are expressions that involve the ratio of two polynomials. In simpler terms, you can think of a rational function as a fraction where both the numerator and the denominator can be polynomials. In the function \( f(x) = \frac{1}{x-3} \), the numerator is \( 1 \), which is a constant polynomial, and the denominator is \( x-3 \), a simple linear polynomial.

Rational functions are interesting because they can have different behaviors depending on the value of \( x \). Since they are defined by a fraction, this means the denominator cannot be zero. Otherwise, the function would be undefined at that point. This is crucial when finding the domain of a rational function, as it denotes all the possible inputs \( x \) can take that won't make the denominator zero.

An essential part of working with rational functions is identifying where the function becomes undefined. This often happens when the denominator is zero. For our example function \( f(x) = \frac{1}{x-3} \), to find where it's undefined, we set the denominator to zero and solve for \( x \), resulting in \( x = 3 \).

Undefined values represent points where the function "breaks down," and they are crucial when determining the function's domain. These values must be excluded from the domain to ensure the function remains valid. So, whenever you encounter a rational function, always look for potential values of \( x \) that can make the denominator zero and thus make the function undefined.

Interval notation provides a convenient way to describe the domain of a function without using lengthy explanations. It succinctly specifies which numbers are included in the domain by denoting start and end points, with special symbols to indicate whether endpoints are included or excluded.

For instance, the domain of \( f(x) = \frac{1}{x-3} \) is all real numbers except for \( x = 3 \). Using interval notation, this is expressed as \( (-\infty, 3) \cup (3, \infty) \). This tells us:

• The function includes all values less than 3, \( (-\infty, 3) \).
• And all values greater than 3, \( (3, \infty) \).

The \( \cup \) symbol combines these two intervals, reinforcing that every number except 3 is a valid input for the function.