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Understanding the domain of a function is crucial when dealing with rational functions. The domain consists of all the input values (x-values) for which the function is defined. In simpler terms, these are the numbers you can safely plug into the function without causing any mathematical havoc.

Let’s consider the function provided: \( F(x) = \frac{4}{x-3} \). Here, the expression becomes undefined when the denominator is zero. That's what you need to inspect:

• Check when the denominator equals zero: \( x - 3 = 0 \).
• This leads to \( x = 3 \).

Therefore, you cannot use \( x = 3 \) in this function, because dividing by zero is impossible. Thus, the domain of the function is all real numbers except 3.

To express the domain, you can write: all real numbers \( x \), such that \( x eq 3 \). This means every value except 3 can be used in the function.

Vertical asymptotes are lines where the graph of a function tends towards infinity. Technically, a vertical asymptote is a vertical line \( x = a \), where the function grows wild as \( x \) approaches \( a \).

For the function \( F(x) = \frac{4}{x-3} \), we identify the vertical asymptote by setting the denominator equal to zero. This is because these x-values make the function undefined.

• Set the denominator to zero: \( x - 3 = 0 \).
• Solve for \( x \) to find that \( x = 3 \).

In this case, \( x = 3 \) is where the vertical asymptote occurs. As the graph approaches this line from either side, the function moves towards positive or negative infinity, creating a barrier that the graph does not cross. Vertical asymptotes are crucial in picturing the behavior of a rational function.

Horizontal asymptotes describe the behavior of a function as the input \( x \) stretches towards infinite values. Essentially, they tell us the y-value the function approaches as \( x \) goes far left or right on the graph.

For \( F(x) = \frac{4}{x-3} \), identifying whether there is a horizontal asymptote involves comparing the degree of the polynomial in the numerator to the degree of the polynomial in the denominator:

• If the numerator's degree is less than the denominator's, the x-axis or \( y = 0 \) is your horizontal asymptote.
• Here, in \( \frac{4}{x-3} \), the degree of the numerator is 0 (since it's just 4), and the denominator’s degree is 1 (since it's \( x \) to the first power).

Since the degree of the numerator is lesser, our horizontal asymptote is \( y = 0 \). This means, as \( x \) stretches to either positive or negative infinity, the graph of \( F(x) \) will tend closer and closer to the line \( y = 0 \), without ever really touching it.