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A rational function is defined as a function that is the ratio of two polynomials. It is written in the form \( F(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not allowed to be zero since division by zero is undefined.

Rational functions often exhibit interesting behaviors near certain values of \( x \) which include vertical asymptotes, horizontal asymptotes, or holes. Vertical asymptotes are particularly significant as they signify points on the graph where the function value grows indefinitely large or drops indefinitely large as one approaches these points from either side.

Identifying vertical asymptotes involves finding values for \( x \) that make the denominator \( Q(x) \) equal to zero, as long as the numerator \( P(x) \) does not also become zero at those same points (in which case you may have a hole instead). Recognizing and understanding these characteristics are essential in analyzing and sketching the graph of a rational function.

Solving equations is foundational in mathematics, particularly when dealing with rational functions and their properties such as vertical asymptotes. The process of finding the values of the variable that satisfy an equation often involves isolating the variable using algebraic manipulations.

In the given exercise, we are tasked with finding the values of \( x \) for which the denominator of our rational function \( F(x) = \frac{3x-5}{x^3-8} \) becomes zero. The equation to solve here is \( x^3 - 8 = 0 \) which simplifies to \( x^3 = 8 \) after rearranging terms. By applying the cube root to both sides, we determine that \( x = 2 \) is the root. It's crucial to recognize that depending on the degree of the polynomial, an equation may have multiple solutions, and all must be examined to identify potential vertical asymptotes.

Graph behavior in rational functions can tell us a lot about its properties, such as continuity, limits, and asymptotic behavior. Vertical asymptotes signify where the function grows unbounded, which on a graph is visualized as the function approaching a line (the asymptote) but never actually touching it.

When graphing rational functions, it is important to locate the vertical asymptotes first, as they serve as important guides to the overall shape and trajectory of the graph. In addition to asymptotes, it's also important to determine whether the function approaches positive or negative infinity as it gets closer to these lines. This is known as the end behavior. Other aspects like intercepts, symmetry, and domain restrictions further contribute to the complete picture of graph behavior of rational functions.

Understanding these concepts helps students accurately predict and sketch the path of a rational function just by examining its equation, before even plotting a single point. This understanding is vital to mastering the concept of graphing in mathematics.