Problem 21 - Algebra and Trigonometry

Question

Find all horizontal and vertical asymptotes (if any). $$ r(x)=\frac{5}{x-2} $$

Step-by-Step Solution

Answer

Vertical asymptote at \(x = 2\) and horizontal asymptote at \(y = 0\).

Step 1: Identify the vertical asymptote

A vertical asymptote may occur where a fraction is undefined. For the function \( r(x) = \frac{5}{x-2} \), the denominator is \(0\) when \(x - 2 = 0\). Solving for \(x\), we find \(x = 2\). Thus, there is a vertical asymptote at \(x = 2\).

Step 2: Identify the horizontal asymptote

For a rational function \( \frac{f(x)}{g(x)} \), where \(f(x)\) and \(g(x)\) are polynomials, the horizontal asymptote depends on the degrees of the numerator and the denominator. Here, \(f(x) = 5\) is a constant (degree 0), and \(g(x) = x - 2\) has degree 1. Because the denominator's degree (1) is greater than the numerator's degree (0), \(y = 0\) is the horizontal asymptote.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes of a function give us an idea of how the function behaves as the input grows very large or very small, approaching infinity or negative infinity. For rational functions, which are quotients of two polynomials, finding horizontal asymptotes involves comparing the degrees of the polynomials. The rules for determining horizontal asymptotes are simple:

• If the degree of the numerator polynomial is less than the degree of the denominator polynomial, the horizontal asymptote is at the line \( y = 0 \).
• If the degrees of the numerator and denominator are equal, the horizontal asymptote is at \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of the numerator and denominator, respectively.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

In the function \( r(x) = \frac{5}{x-2} \), the numerator (\( 5 \)) is a constant, which means its degree is 0. The denominator, \( x - 2 \), has a degree of 1. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at \( y = 0 \). This tells us that as \( x \) becomes very large or very small, the values of \( r(x) \) will approach 0.

Vertical Asymptotes

Vertical asymptotes occur where a function approaches infinity or negative infinity as the input approaches a particular point. For rational functions, these asymptotes are found where the denominator becomes 0, causing the function to be undefined at that point. To find vertical asymptotes, we:

• Set the denominator equal to 0 and solve for \( x \).
• Ensure that the numerator does not also become 0 for these \( x \)-values, as this would indicate a hole instead of an asymptote.

For \( r(x) = \frac{5}{x-2} \), the denominator is \( x - 2 \). Setting it to zero gives us \( x - 2 = 0 \), or \( x = 2 \). At \( x = 2 \), the function is undefined, and there is no cancellation with the numerator (since the numerator is a constant and non-zero everywhere). Thus, we have a vertical asymptote at \( x = 2 \). This means as \( x \) approaches 2 from either direction, the function's value will go to infinity or negative infinity.

Rational Functions

Rational functions are expressions that can be written as the ratio of two polynomials, in the form \( \frac{f(x)}{g(x)} \), where \( f(x) \) is the numerator and \( g(x) \) is the denominator. Understanding rational functions is key, as their behavior and properties are often influenced by the degrees and coefficients of these polynomials.### Key Features of Rational Functions- **Domain:** The domain of a rational function is all real numbers except where the denominator is zero. These points often relate to vertical asymptotes or holes.- **Asymptotes:** Rational functions may have both horizontal and vertical asymptotes, describing their behavior at infinity and specific points, respectively.### Asymptotes Insight- **Vertical Asymptotes:** Occur at zeros of the denominator, not canceled by the numerator.- **Horizontal/Oblique Asymptotes:** Determined by the degree and leading coefficients of the numerator and denominator.Rational functions are fascinating due to their ability to model a wide variety of behaviors seen in real-world applications, from physics to economics. By understanding how to identify and analyze asymptotes, we can predict the function's behavior in different contexts.