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Vertical asymptotes are lines that the graph of a function approaches but never touches as the inputs either increase or decrease without bound. In mathematical terms, if a function f(x) has a vertical asymptote at x = a, then as x approaches a from the right or left, the function's values approach infinity or negative infinity.

In our exercise, the vertical asymptotes are identified by finding the values for x that make the denominator zero, as the function cannot be defined at these points. In the given function, the denominator is zero when x = 3 or x = -3. So, the graph of the function will have vertical asymptotes at these x-values. These lines, represented by the equations x = 3 and x = -3, are crucial to consider when sketching the graph, as they partition the coordinate plane and guide the shape of the graph near these x-values.

Horizontal asymptotes represent the value that a function's output (y-value) approaches as the input (x-value) goes to positive or negative infinity. When you have a rational function, the horizontal asymptote can be determined by comparing the degrees of the polynomials in the numerator and denominator.

For the function f(x) = \(\frac{x+3}{x^{2}-9}\), we see that the degree of the denominator (2) is greater than the degree of the numerator (1), which means the y-values of the graph approach 0 as x approaches infinity in either direction. Therefore, the function has a horizontal asymptote at y = 0. This asymptote serves as a boundary that the function will get closer and closer to as the absolute value of x becomes very large, but it will never cross.

A rational function is a function that can be expressed as the ratio of two polynomials. It is written in the form f(x) = \(\frac{P(x)}{Q(x)}\), where both P(x) and Q(x) are polynomials, and Q(x) is not the zero polynomial. The properties of a rational function's graph can be complex and influenced by factors including the degrees of the numerator and denominator, zeros of the polynomials, and the presence of asymptotes.

The function given in the exercise, \(f(x) = \frac{x+3}{x^2-9}\), is an example of a rational function with vertical asymptotes and a horizontal asymptote. Its graph shows that it behaves differently in different regions of the plane, defined by these asymptotes. Understanding the characteristics of rational functions helps us predict the shape and direction of their graphs and determine their long-term behavior.