A rational function is essentially a fraction where the numerator and denominator are both polynomials. Think of something like $f(x) = \frac{x+5}{x-2}$. Just like with simple fractions, there's one critical rule we can never break: the denominator cannot be zero.

The set of all possible $x$-values a function can accept is called its domain. For rational functions, our main task is to find which $x$-values would break the 'no zero in the denominator' rule and exclude them. These exclusions are called points of discontinuity. What's left over is the function's domain.

The process is straightforward. First, isolate the polynomial in the denominator. Second, set that polynomial as not equal to zero. Finally, solve the resulting equation. The solutions are your excluded values.

Let's start with a simple case where the denominator is linear. Consider the function $f(x) = \frac{2x+1}{x+4}$.

Things get more interesting when the denominator is a quadratic polynomial. We often need to factor it first to find the roots.

Consider $g(x) = \frac{x}{x^2 - 5x + 6}$.

For the product of two terms to be non-zero, neither term can be zero. This is the inverse of the zero product property. This gives us two separate conditions.

Once we have the excluded values, it's helpful to visualize the domain on a number line. We simply draw a line and mark the excluded points with open circles. These open circles signify that the value is not included in the domain.

The shaded regions on the number line show us the valid parts of the domain. This visual representation leads directly to a formal mathematical description called interval notation., an annotation on 'interval notation' is a good idea. This is a technical term that warrants a deep dive.

For our function $g(x)$ with exclusions at 2 and 3, the number line is broken into three pieces. We write the domain in interval notation like this:

The same process applies no matter how many excluded values you have. A denominator like $x(x+1)(x-5) eq 0$ would simply result in three excluded points ($0, -1, 5$) and four separate intervals in the domain.