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The domain of a rational function refers to all the possible values of \(x\) for which the function is defined. For the function \(f(x) = \frac{4-2x}{3x-1}\), the denominator plays a crucial role. A function is undefined wherever its denominator is zero, so we must ensure the denominator is never zero for the domain.
*Finding the Domain*
To find when a rational function is undefined, set the denominator equal to zero and solve for \(x\).

• For \(3x - 1 = 0\), solve this to get \(x = \frac{1}{3}\).

As a result, the domain includes all real numbers except \(x = \frac{1}{3}\). This approach helps in determining the domain of any rational function and provides insight into where the function behaves anomalously.

Vertical asymptotes are lines that the graph of a function approaches but never touches or crosses. They occur at those points where the function is undefined due to the denominator being zero, but the numerator is not zero.
*Locating Vertical Asymptotes*
Recall from the domain that the vertical asymptote appears exactly at points where the denominator leads to zero:

• In this case, setting \(3x - 1 = 0\) led us to \(x = \frac{1}{3}\).

Thus, there is a vertical asymptote at \(x = \frac{1}{3}\). It is essential to recognize these characteristics in graphs as they indicate a crucial behavior change in the function's proximity.

Horizontal asymptotes describe the behavior of a function as \(x\) approaches infinity or negative infinity. They help in understanding the end behavior of a graph. For a rational function, the degrees of the polynomials in the numerator and denominator determine the presence of horizontal asymptotes.
*Determining Horizontal Asymptotes*
Examine the degrees:

• The numerator \(4 - 2x\) and the denominator \(3x - 1\) are both of degree 1.

For these cases, the horizontal asymptote is the ratio of the leading coefficients. For \(\frac{-2}{3}\), the leading coefficient of the numerator is \(-2\), and for the denominator, it is \(3\). So the horizontal asymptote here is \(y = \frac{-2}{3}\). This makes it apparent how the graph levels out over long stretches of \(x\) values.