Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. $$ f(x)=\frac{2}{5 x+2} $$

Step-by-Step Solution

Answer

Domain: all real numbers except \( x = -\frac{2}{5} \); vertical asymptote: \( x = -\frac{2}{5} \); horizontal asymptote: \( y = 0 \).

Step 1: Find the Domain

To find the domain of the function \( f(x) = \frac{2}{5x + 2} \), we need to identify any values of \( x \) that would make the denominator zero since division by zero is undefined. Set the denominator equal to zero and solve for \( x \):\[ 5x + 2 = 0 \]Subtract 2 from both sides:\[ 5x = -2 \]Divide both sides by 5:\[ x = -\frac{2}{5} \]This means \( x = -\frac{2}{5} \) is not in the domain. Therefore, the domain is all real numbers except \( x = -\frac{2}{5} \).

Step 2: Analyze for Vertical Asymptotes

Vertical asymptotes occur at the values of \( x \) that make the denominator zero, as these are points where the function can become undefined. From Step 1, we have already determined that \( x = -\frac{2}{5} \) makes the denominator zero. Therefore, there is a vertical asymptote at \( x = -\frac{2}{5} \).

Step 3: Analyze for Horizontal Asymptotes

To find horizontal asymptotes, we compare the degrees of the polynomial in the numerator and the denominator of the function \( f(x) = \frac{2}{5x + 2} \). The numerator is a constant (degree 0), and the denominator is linear (degree 1). When the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is \( y = 0 \), which means the function approaches the x-axis as \( x \) goes to infinity or negative infinity.

Key Concepts

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. Just like fractions express ratios of integers, rational functions represent ratios involving variables. This type of function can be written as \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \).
Understanding rational functions involves knowing their behavior and identifying key characteristics like domain, asymptotes, and intercepts.

• The domain is determined by setting the denominator \( Q(x) \) not equal to zero, to avoid division by zero.
• Asymptotes are lines that the graph approaches but never touches, representing the function's behavior at its extremes.

Mastering these aspects will make it easier to sketch the graph of a rational function and predict its behavior as the variable \( x \) increases or decreases.

Vertical Asymptotes

Vertical asymptotes are a fundamental feature of rational functions. They occur at the values of \( x \) that cause the denominator to be zero while the numerator is non-zero. This is because the function tends towards infinity at these points, thus creating a vertical line that the function cannot cross.
To find vertical asymptotes, set the denominator equal to zero and solve for \( x \). For example, in the function \( f(x) = \frac{2}{5x + 2} \), we solve:
\[5x + 2 = 0\]
\[x = -\frac{2}{5}\]
Hence, a vertical asymptote exists at \( x = -\frac{2}{5} \).

• Vertical asymptotes indicate where a function is undefined due to division by zero.
• They often separate the graph into distinct behaviors on either side, showing how the function shoots up to positive or down to negative infinity near these lines.

Recognizing these asymptotes is crucial for graphing and understanding the function's limits.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a rational function as \( x \) approaches infinity or negative infinity. They are horizontal lines corresponding to a constant \( y \)-value that the function approaches but never touches.
To determine horizontal asymptotes, compare the degrees of the numerator and denominator polynomial.
For the function \( f(x) = \frac{2}{5x + 2} \):

• The degree of the numerator is 0 (it's a constant \( 2 \)).
• The degree of the denominator is 1 (it's linear \( 5x + 2 \)).

In this case, since the degree of the denominator is greater than that of the numerator, the horizontal asymptote is \( y = 0 \).

• An asymptote of \( y = 0 \) indicates the function approaches the x-axis as \( x \) increases or decreases indefinitely.
• Horizontal asymptotes are key in understanding end-behavior and the limiting value of the function.

Keep in mind that while functions approach, they may never touch a horizontal asymptote, but they provide critical information on long-range behavior.