91Ó°ÊÓ

Rational functions are mathematical expressions that involve ratios of polynomials. Simply put, they are fractions where the numerator and denominator are polynomials. The domain of a rational function is essential to consider as it determines all the possible input values (or "x" values) for which the function is defined.
Understanding the domain involves ensuring the denominator does not equal zero, because division by zero is undefined in mathematics. In the given exercise, the function is \( f(x) = \frac{x+7}{x^2+49} \). To find the domain, we focus on the denominator \( x^2 + 49 \) and solve \( x^2 + 49 = 0 \). If a solution exists, those values must be excluded from the domain. However, as you will discover, there are no real solutions to this equation, indicating a specific case when determining domains.

Real numbers are familiar quantities that encompass all the numbers on the number line. They include positive numbers, negative numbers, zero, and even fractions and irrational numbers. Real numbers do not, however, include imaginary numbers.
The set of real numbers is important when discussing domains of rational functions because these are the inputs for which functions are initially evaluated. In our particular exercise, when we attempt to solve \( x^2 = -49 \) for real numbers, we are seeking numbers that could make the denominator zero, affecting the function's domain. The real numbers cannot include the solutions \( x = \pm \sqrt{-49} \) as no real number when squared results in a negative number. Therefore, the value does not exist in the real number system, meaning all real numbers are permissible for the function input in this case.

Imaginary numbers extend our number system to solve equations that real numbers cannot. They involve the square root of negative numbers, represented with the symbol "i," where \( i^2 = -1 \).
In the exercise, solving \( x^2 = -49 \) leads us to an imaginary number, as \( x = \pm 7i \). Imaginary numbers help express roots of negative numbers where real solutions are not possible.
Though imaginary numbers are crucial in advanced mathematical contexts, especially in complex analysis, they do not influence the domain of rational functions over real numbers. Hence, in problems concerning the real-world applications of rational functions, we often only consider real number domains and exclude imaginary solutions.