91Ó°ÊÓ

When analyzing a rational function, understanding the structure of the denominator is crucial. A rational function is essentially one polynomial divided by another. The denominator determines the values that are allowed in the function. We need to ensure that the denominator never equals zero because division by zero is undefined.

In the exercise, the denominator of the rational function is given as \(c^2 + 6\). We analyze this expression to determine if there are any values of \(c\) that make it zero. A zero denominator would mean that the function is not defined for those particular values, and we would have to exclude them from the domain.

The next step involves solving equations that derive from setting the denominator equal to zero. This is a technique to find more about which values need to be excluded from the domain.

For the exercise, we take the denominator \(c^2 + 6\) and set up the equation:

• \(c^2 + 6 = 0\)

To solve it, subtract 6 from both sides to get \(c^2 = -6\). Notice that real numbers squared cannot result in a negative number. Therefore, there are no real solutions to this equation. This indicates that the denominator never reaches zero for any real number, implying that no real value of \(c\) needs to be excluded from the domain.

Rational functions are a fundamental concept in algebra and calculus. They are expressed as the quotient of two polynomials. Understanding their properties, especially the domain, is essential.

The function in the exercise, \(A(c)=\frac{8}{c^2+6}\), illustrates the simplicity of working with rational functions when the denominator does not include terms that lead to division by zero. Since \(c^2 + 6\) does not equal zero for real \(c\), it simplifies our task to determine the domain. Every value that can be submitted without making the denominator zero is part of the domain of a rational function.

The solution highlights the importance of understanding the set of real numbers in mathematics. Real numbers include all the numbers on the number line, embracing both the rational and irrational numbers.

In the context of rational functions, especially when analyzing the domain, we concern ourselves with whether there are real-number solutions to equations like \(c^2 + 6 = 0\). Since there are none (given that \(c^2\) must be non-negative), the domain remains all real numbers, indicating that the function is defined over the entire set of real numbers. This results in the domain being \((-\infty, \infty)\), which is a straightforward indication of a function without restrictions for real number inputs.