Problem 78 - Algebra 2

Question

Find the vertical asymptotes of each function. $$ y=\frac{x-3}{x+1} $$

Step-by-Step Solution

Answer

The vertical asymptote of the function \( y=\frac{x-3}{x+1} \) is \( x=-1 \).

Step 1: Identify the Denominator

The denominator of the given function \(y=\frac{x-3}{x+1}\) is \(x+1\). This is the expression we'll set equal to zero.

Step 2: Set Denominator Equal to Zero

Setting the denominator equal to zero gives us the equation, \(x+1=0\).

Step 3: Solve for x

To solve for x, we subtract 1 from both sides of the equation, \(x+1-1=0-1\). This simplifies to \(x=-1\).

Key Concepts

Understanding Rational Functions

A rational function is a type of mathematical function, represented as a fraction, with a polynomial in both the numerator and denominator. In simpler terms, it looks like \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. The function deals with two components:

• **Numerator**: The top part of the fraction.
• **Denominator**: The bottom part of the fraction.

In the expression \( y = \frac{x-3}{x+1} \), \( x-3 \) is the numerator and \( x+1 \) is the denominator. Rational functions are important because they can show patterns in data and are often used in calculus and algebra to understand behavior patterns around their asymptotes and intercepts.

The Role of the Denominator

The denominator, in particular, plays a crucial role in understanding and working with rational functions. It is important because it indicates where the function is undefined. Since division by zero is not possible, any values that make the denominator zero need to be identified and excluded from the domain of the function.

For the function \( y = \frac{x-3}{x+1} \), the denominator is \( x+1 \). To find when the function is undefined, we must find the value of \( x \) that makes \( x+1 = 0 \). This is why the denominator is set to zero in our calculations. When we solve this equation, it helps us identify potential points of discontinuity such as vertical asymptotes.

Exploring Asymptotes

Asymptotes are lines that a graph approaches but never actually reaches. In rational functions, vertical asymptotes are particularly significant as they mark the values that the function can never take on the x-axis, resulting in undefined points.

In our given function \( y = \frac{x-3}{x+1} \), plugging in \( x = -1 \) into the denominator results in zero, suggesting a vertical asymptote at this point. Graphically, as \( x \) approaches -1 from either direction (left or right), the values of \( y \) tend to increase or decrease without bound, meaning it shoots upwards or downwards indefinitely.

Vertical asymptotes are useful in analyzing the behavior of graphs of rational functions, helping to understand their tendency near certain points.

Steps in Solving Equations

Solving equations, especially in the context of identifying vertical asymptotes in rational functions, often involves a series of methodical steps:

• **Identify the denominator.**: Focus on the bottom part of the fraction (e.g., \( x+1 \) in our example).
• **Set the denominator equal to zero.**: This helps identify values that make the function undefined (e.g., \( x+1 = 0 \)).
• **Solve for \( x \).**: Determine the specific \( x \) values that satisfy the equation. For instance, solve \( x+1 = 0 \) by subtracting 1, getting \( x = -1 \).

This process is valuable because it tells us where the vertical asymptotes are, ensuring a complete understanding of the function's behavior at and around these critical points.