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When you need to describe the domain of a function, you are essentially identifying all the x-values for which the function is defined. In cases where a function isn't defined for particular x-values, as is the case with rational functions where the denominator cannot be zero, the domain is often expressed as a union of intervals. Imagine a number line. There might be certain points where the function does not work. These are usually noted as open intervals, which are segments of the number line that do not include the points where the function is undefined. For example, if a function has undefined points at \(x = a\) and \(x = b\), the domain could be expressed as the union of intervals \((-\infty, a) \cup (a, b) \cup (b, +\infty)\). This covers all segments of the number line, skipping over where the function breaks. Here's a simple breakdown:- Each "interval" includes a segment where the function is defined.- "Union" symbol \( \cup \) combines all such intervals into a single domain expression.This approach clearly communicates which x-values are valid inputs for the function.

A rational function is a type of function that is the ratio of two polynomials. This means that there is a polynomial in the numerator and another in the denominator. The general form is \(r(x) = \frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials. These functions are called "rational" just like the term "rational number," which is the ratio of two integers. A common characteristic of rational functions is that they can be undefined for some x-values—specifically, where the denominator equals zero, leading to division by zero, which is undefined. In the example \( r(x) = \frac{5x^3 - 12x^2 + 13}{x^2 - 7} \), the numerator is \(5x^3 - 12x^2 + 13\) and the denominator is \(x^2 - 7\). Solving for when the denominator equals zero helps us determine the points to exclude from the domain.

Finding the roots of the denominator in a rational function is essential for identifying points where the function is not defined and thus determining the domain. To find the denominator roots, set the denominator equal to zero and solve for \(x\). In the given function, \(r(x) = \frac{5x^3 - 12x^2 + 13}{x^2 - 7}\), the denominator is \(x^2 - 7\). By setting \(x^2 - 7 = 0\), we solve:- Move 7 to the other side: \(x^2 = 7\)- Take the square root of both sides: \(x = \pm \sqrt{7}\)These values, \(x = \sqrt{7}\) and \(x = -\sqrt{7}\), are where the function's denominator becomes zero. As a result, they must be excluded from the domain, since at these points the function is undefined. This exclusion results in a gap around these roots on the number line. Understanding how to find and handle these roots is crucial for correctly determining the domain of rational functions.