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A rational function is a mathematical expression representing a ratio of two polynomial functions. In essence, it takes the form \(\frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomial expressions and \(q(x) \eq 0\). For instance, the function involved in our textbook exercise \(h(x)=\frac{12 x}{x^{2}-36}\) falls into this category because it's composed of a linear polynomial in the numerator and a quadratic polynomial in the denominator.

It's crucial to understand that the domain of a rational function is all the real numbers for which the denominator is non-zero, because any value of \(x\) making the denominator zero would make the entire expression undefined. That is, we cannot have division by zero in mathematics. Identifying the domain involves finding the values for which the denominator \(q(x)\) equals zero and excluding those from the set of all real numbers.

In the universe of mathematical functions, there are circumstances under which a function does not yield a value, and we describe the function as being undefined for certain inputs. For rational functions, this occurs when the denominator is zero, because division by zero is not a permissible operation in standard mathematics. We must, therefore, be vigilant when dealing with rational functions to avoid these values.

Returning to our example, \(h(x)=\frac{12 x}{x^{2}-36}\), we'd pinpoint the values that make the denominator, \(x^{2}-36\), equal to zero. Solving \(x^{2}-36=0\) gives us \(x=\pm6\). Hence, \(h(x)\) is undefined at \(x=6\) and \(x=-6\). When a student encounters such a scenario, they must exclude these specific undefined points from the domain of the function.

Interval notation is a succinct, mathematically rigorous way to express a range of numbers, typically used to denote the domain or range of a function. A domain composed of all real numbers except some isolated points is represented using union symbols (\(\cup\)) and intervals separated by these excluded points. For rational functions with non-continuous domains, this notation becomes particularly handy.

When we determine the domain of the function \(h(x)=\frac{12 x}{x^{2}-36}\), we exclude the points where \(x=6\) and \(x=-6\). Thus, the domain consists of three separate intervals: from negative infinity to -6, from -6 to 6, and from 6 to positive infinity. These intervals are disjoint because we purposely leave out \(x=6\) and \(x=-6\). This domain is expressed in interval notation as \( (-\infty, -6) \cup (-6, 6) \cup (6, \infty) \), efficiently conveying all the x-values for which the function is defined.