Problem 36 - Calculus

Question

In Problems \(36-38,\) find all horizontal and vertical asymptotes for each rational function. $$f(x)=\frac{5 x-2}{2 x+3}$$

Step-by-Step Solution

Answer

Vertical asymptote: \( x = -\frac{3}{2} \), Horizontal asymptote: \( y = \frac{5}{2} \).

Step 1: Identifying a Rational Function

The given function is a rational function of the form \( \frac{a(x)}{b(x)} \) where \( a(x) = 5x - 2 \) and \( b(x) = 2x + 3 \). Our goal is to find the asymptotes of this function.

Step 2: Finding Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function equals zero. Set \( 2x + 3 = 0 \) and solve for \( x \):\[ 2x + 3 = 0 \]\[ 2x = -3 \]\[ x = -\frac{3}{2} \]This means there is a vertical asymptote at \( x = -\frac{3}{2} \).

Step 3: Finding Horizontal Asymptotes

Horizontal asymptotes are found by analyzing the degrees of the numerator and the denominator. Here, both the numerator (\(5x - 2\)) and the denominator (\(2x + 3\)) are of degree 1. For rational functions where the degree of the numerator and denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients. Therefore:\[ y = \frac{5}{2} \]This means there is a horizontal asymptote at \( y = \frac{5}{2} \).

Key Concepts

Rational Functions

A rational function is a type of function that is expressed as the quotient of two polynomials, written in the form \( \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials, and \( q(x) eq 0 \). This means that your rational function can look like a fraction made up of two polynomial expressions. For example, in the function \( f(x) = \frac{5x - 2}{2x + 3} \), \( 5x - 2 \) functions as the numerator, while \( 2x + 3 \) is the denominator.
Keep in mind rational functions can model many real-world scenarios like rates, proportions, and other changes that aren't constant. Learning how to find their asymptotes can help you understand the behavior of the function over time or space.

• Make sure the denominator is never zero, as dividing by zero is undefined.
• The degrees of the numerator and the denominator are crucial to determining the types of asymptotes.

Vertical Asymptotes

Vertical asymptotes are imaginary lines that the graph of a rational function will approach but never actually reach or cross, indicating a vertical boundary in terms of values of \( x \). These occur at the values where the denominator equals zero and thus the function is undefined.
To find vertical asymptotes in a rational function, you'll need to solve for \( x \) by setting the denominator equal to zero:
For \( f(x) = \frac{5x - 2}{2x + 3} \), set \( 2x + 3 = 0 \). Solving this gives \( x = -\frac{3}{2} \), which means there is a vertical asymptote at \( x = -\frac{3}{2} \).
Remember, rational functions can have multiple vertical asymptotes depending on how many roots the denominator has. But they can't have them if the factor cancels out with the numerator, as that signifies a hole rather than an asymptote.

Horizontal Asymptotes

Horizontal asymptotes are horizontal lines that the graph of a rational function approaches as \( x \) becomes very large or very small, indicative of how the function behaves at infinity.
The presence and position of horizontal asymptotes depend on the degrees of the polynomials in the rational function:

• If the degree of the numerator is less than that of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degree of the numerator equals the degree of the denominator, as in \( f(x) = \frac{5x - 2}{2x + 3} \), the horizontal asymptote is the ratio of the leading coefficients. Here, this ratio is \( \frac{5}{2} \), so \( y = \frac{5}{2} \).
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote; instead, the function may have an oblique (slant) asymptote.

Horizontal asymptotes describe the end behavior of a function and help us understand the function's limits as \( x \) approaches positive or negative infinity.