Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. $$ f(x)=\frac{4-2 x}{3 x-1} $$

Step-by-Step Solution

Answer

Domain: \(x \neq \frac{1}{3}\). Vertical asymptote: \(x = \frac{1}{3}\). Horizontal asymptote: \(y = -\frac{2}{3}\).

Step 1: Find Domain

To find the domain of the function \(f(x) = \frac{4-2x}{3x-1}\), we need to determine the values of \(x\) for which the function is defined. The function is undefined where the denominator is equal to zero. So, we solve the equation \(3x - 1 = 0\) to find these values. \(3x = 1\) yields \(x = \frac{1}{3}\). Therefore, the domain is all real numbers except \(x = \frac{1}{3}\).

Step 2: Identify Vertical Asymptotes

Vertical asymptotes occur where the function is undefined, and can be found by setting the denominator equal to zero. From Step 1, we found that \(3x - 1 = 0\) leads to \(x = \frac{1}{3}\). Therefore, the vertical asymptote is at \(x = \frac{1}{3}\).

Step 3: Find Horizontal Asymptotes

Horizontal asymptotes are determined by the degree of the polynomial in the numerator and the denominator. In this function, both the numerator \(4 - 2x\) and the denominator \(3x - 1\) are linear polynomials (degree 1). When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. Therefore, the horizontal asymptote is \(y = \frac{-2}{3}\).

Key Concepts

Domain of a Function

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

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

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.