91Ó°ÊÓ

Rational functions are mathematical expressions represented as the ratio of two polynomials. The general form of a rational function is \( f(x) = \frac{p(x)}{q(x)}\) where \( p(x)\) and \( q(x)\) are polynomials and \( q(x) \eq 0\).

The domain of a rational function includes all real numbers except those that make the denominator zero. In the given exercise, \( f(x)=\frac{x+8}{x^{2}+64} \), determining the domain involves examining the denominator \(x^{2}+64\) for values of \(x\) that might make it zero. Since the denominator cannot be zero, the values that solve \(x^{2} + 64 = 0\) are excluded from the domain.

Understanding the domain is crucial as it tells us the set of all possible inputs for the function, ensuring that we do not encounter undefined expressions when working with rational functions.

Complex roots arise when solving equations that do not have solutions in the set of real numbers. They are expressed in the form \( a + bi \) where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, satisfying \( i^{2} = -1 \).

In reference to the exercise at hand, the quadratic equation \(x^{2} + 64 = 0\) yields a pair of complex roots, \(x = \pm 8i\). These roots are not part of the domain of the rational function because the function is defined over the real numbers only. This illustrates how the concept of complex roots complements our understanding of the domain, indicating values that should be excluded even though they don't directly influence the real number domain.

The Quadratic Formula

Quadratic equations are polynomial equations of the second degree, generally written as \( ax^2 + bx + c = 0 \). They can be solved by various methods, including factoring, completing the square, and using the quadratic formula
\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \].

In this exercise, we have a quadratic equation \(x^{2} + 64 = 0\). Since it is already in standard form and cannot be factorized easily, we move on to find its roots using the relevant methods. Finding the roots of the quadratic equation is a fundamental step to determine the values to exclude from the domain of the rational function.

Real numbers are the set of numbers that include both rational and irrational numbers, covering every point on the number line. In the context of the domain of rational functions, only real numbers are typically considered unless otherwise specified.

When we state that the domain consists of all real numbers barring those that make the denominator zero, we are inherently excluding complex roots, non-real solutions. For the exercise given, while the quadratic equation presents complex roots, these are not recognized within the domain, emphasizing that the function \( f(x)\) is grounded in real numbers and specifics its domain as such.