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A rational function is a type of function that can be expressed as a fraction where both the numerator and the denominator are polynomials. These functions are quite useful in modeling real-world scenarios that have a ratio or rate component, such as speed (distance over time) and density (mass over volume). They take the form:\[f(x) = \frac{P(x)}{Q(x)}\]where \(P(x)\) and \(Q(x)\) are polynomials. Key points of rational functions include:

• Numerator: The top part of the fraction, which can be any polynomial, even just a constant number.
• Denominator: The bottom part of the fraction, itself a polynomial, but crucially it cannot be zero.

This brings us to a critical consideration regarding the domain of rational functions. Since division by zero is undefined in mathematics, the values that make the denominator zero are excluded from the domain.

The denominator plays a crucial role in defining the domain of rational functions. If the denominator is zero, the rational function becomes undefined. Therefore, identifying when the denominator equals zero is essential in determining the domain of the function. Let's look at the function given in the exercise:\[f(x) = \frac{7x}{x-8}\]Here, the denominator is \(x - 8\). This means the function is undefined when \(x - 8 = 0\), since it would entail division by zero.To find out when this occurs:1. Set the denominator equal to zero: \(x - 8 = 0\)2. Solve the equation: \(x = 8\)This result indicates that when \(x\) equals 8, the denominator is zero. Thus, \(x = 8\) is excluded from the domain of the function.

Real numbers encompass all the numbers on the continuous number line, including both rational numbers (like 1/2 or 7) and irrational numbers (like \(\sqrt{2}\)). In mathematics and rational functions, the domain is often expressed in terms of real numbers.For the rational function \(f(x) = \frac{7x}{x-8}\), the domain is all real numbers, with a specific exception. Besides the value discovered that makes the function undefined, every other real number can be part of the domain.Let’s consider:

• Real numbers include any value such as fractions, integers, and decimals.
• Real numbers can be both positive and negative, or even zero, unless specifically excluded.

Thus, for this function, any real number is appropriate except \(x = 8\), which means the domain is all real numbers \(\mathbf{except}\) \(8\). This ensures that the rational function remains defined and valid across its entire applicable domain.