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Rational functions are fractions where both the numerator and the denominator are polynomials. For example, in the function \( f(x) = \frac{8}{3x-6} \), the numerator is 8 (which is a constant polynomial), and the denominator is \(3x-6\). Rational functions are interesting because they are defined by where their denominators are not zero. Anytime the denominator equals zero, the function becomes undefined because division by zero is not possible.
In general, rational functions are used to model a variety of situations in mathematics and applied sciences, such as rates of change, and have properties that can make them more complex than simple linear or polynomial functions.
To better understand rational functions, it's key to analyze their form and how their variables interact in the numerator and denominator.

When dealing with rational functions, one of the primary tasks is identifying where the function is undefined. This happens when the denominator of the fraction equals zero. For the function \( f(x) = \frac{8}{3x-6} \), we must find the value of \( x \) that makes \( 3x-6 \) equal to zero.
Here's how we can do it in steps:

• Set the denominator not equal to zero: \(3x - 6 eq 0\).
• Solve the equation: Add 6 to both sides to get \(3x = 6\).
• Divide by 3 to isolate \( x \): \( x = 2\).

This indicates that the function is undefined at \( x = 2 \). Any value of \( x \) except 2 is allowed because only at \( x = 2 \) does the denominator become zero, leading to an undefined function.

Once we've identified the point at which the function is undefined, we can express the domain of the function using interval notation. Interval notation helps describe continuous sets of numbers without writing them out individually.
For the function \( f(x) = \frac{8}{3x-6} \), since it is undefined at \( x = 2 \), we include all real numbers except 2. This is shown as:
\[(-fty, 2) \cup (2, fty)\]
Breaking it down, it means:

• \((-\infty, 2)\) includes all numbers less than 2.
• \((2, fty)\) includes all numbers greater than 2.

By using \( \cup \) (union symbol), we combine these two intervals to form a complete domain that excludes the point where the function is undefined, making it easy to see the range of valid inputs for the function.