Problem 22 - Algebra and Trigonometry

Question

Find all horizontal and vertical asymptotes (if any). $$ r(x)=\frac{2 x-3}{x^{2}-1} $$

Step-by-Step Solution

Answer

Vertical asymptotes at \(x = 1\) and \(x = -1\); horizontal asymptote at \(y = 0\).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur when the denominator of the rational function is equal to zero and the numerator is not equal to zero. We need to solve the equation \(x^2 - 1 = 0\). This factors into \((x - 1)(x + 1) = 0\). Therefore, the solutions are \(x = 1\) and \(x = -1\). So, there are vertical asymptotes at \(x = 1\) and \(x = -1\).

Step 2: Determine the Horizontal Asymptote

Horizontal asymptotes are found by comparing the degrees of the numerator and the denominator. The degree of the numerator, \(2x - 3\), is 1, and the degree of the denominator, \(x^2 - 1\), is 2. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).

Step 3: Check for Simplification

Check if the fraction \(r(x) = \frac{2x - 3}{x^2 - 1}\) can be simplified or canceled further. Since \(2x - 3\) does not factor to include \(x - 1\) or \(x + 1\), no simplification or cancellation is possible here.

Key Concepts

Understanding Vertical Asymptotes

Vertical asymptotes help understand the behavior of a rational function as it approaches certain values of the independent variable. In context to rational functions, they occur at the values of \( x \) where the denominator turns zero and the numerator remains non-zero.
This situation causes the function's value to increase or decrease without bound, hence depicting a sort of 'boundary' in the graph of the function.To find them, take the denominator of the function \( r(x) = \frac{2x-3}{x^2-1} \) and set it equal to zero.
Factor the expression and solve for \( x \):

• Start with \( x^2 - 1 = 0 \).
• Factor it into \( (x - 1)(x + 1) = 0 \).
• This gives \( x = 1 \) and \( x = -1 \) as the values causing vertical asymptotes.

Therefore, the graph of this function will have a vertical asymptote at \( x = 1 \) and \( x = -1 \). Guarantee to check that these values do not simultaneously make the numerator zero; thankfully, here they do not, confirming valid vertical asymptotes.

Deciphering Horizontal Asymptotes

Horizontal asymptotes give insight into the end behavior of a function. They depict how the value of a function behaves as \( x \) moves towards positive or negative infinity.
For rational functions, understanding horizontal asymptotes comes down to comparing the degree (highest power of \( x \)) of the numerator and the denominator.In the function \( r(x) = \frac{2x-3}{x^2-1} \):

• The degree of the numerator \( (2x-3) \) is 1.
• The degree of the denominator \( (x^2-1) \) is 2.

Since the degree of the numerator is less than that of the denominator, the horizontal asymptote of this rational function is \( y = 0 \).
This means as \( x \) becomes very large in either positive or negative direction, \( r(x) \) approaches 0.

Grasping Rational Functions

Rational functions are expressions represented as a fraction where both the numerator and the denominator are polynomials. Understanding these functions involves recognizing their components and peculiarities such as asymptotes.In the given function \( r(x) = \frac{2x-3}{x^2-1} \), identifying its behavior relies significantly on knowing its restrictions and simplifications:

• Vertical asymptotes emerge from the points that make the denominator zero, therefore showing restrictions on the \( x \) values.
• Horizontal asymptotes guide us about the value the function approaches as \( x \) grows extremely large or small.

These insights are essential because they describe important characteristics of the rational function’s graph, defining its limits and behavior.
Assess oxidation between variables to ascertain the continuity and discontinuity.