Problem 16 - Algebra 2

Question

Determine the equations of any vertical asymptotes and the values of \(x\) for any holes in the graph of each rational function. $$ f(x)=\frac{x-5}{x^{2}-4 x-5} $$

Step-by-Step Solution

Answer

Hole at \(x=5\); vertical asymptote at \(x=-1\).

Step 1: Factor the Denominator

The first step is to factor the denominator of the function. The denominator is given by the quadratic expression \(x^2 - 4x - 5\). We factor it by looking for two numbers that multiply to \(-5\) and add to \(-4\). These numbers are \(-5\) and \(1\). Thus, the factored form of the denominator is \((x-5)(x+1)\).

Step 2: Identify Potential Vertical Asymptotes and Holes

A rational function has potential vertical asymptotes or holes where its denominator is zero. Set the factored denominator \((x-5)(x+1)\) equal to zero to identify these points: \((x-5) = 0\) or \((x+1) = 0\). Solving these gives potential points at \(x=5\) and \(x=-1\).

Step 3: Determine the Holes and Vertical Asymptotes

Check if any of these points result in a hole instead of a vertical asymptote. A hole occurs if a factor in the denominator also appears in the numerator. In the function \(f(x) = \frac{x-5}{(x-5)(x+1)}\), the numerator \(x-5\) cancels with \(x-5\) in the denominator, indicating a hole at \(x=5\). The remaining factor \(x+1\) implies a vertical asymptote at \(x=-1\).

Key Concepts

Rational Functions

A rational function is a fraction where both the numerator and the denominator are polynomials. In simpler terms, it's like a division problem where the top and bottom parts are mathematical expressions involving powers of \(x\). These types of functions can have interesting features in their graphs, such as vertical asymptotes, holes, and horizontal asymptotes. To understand rational functions, consider this example:

• \(f(x) = \frac{x-5}{x^2 - 4x - 5}\)

Here, \(x - 5\) is the numerator, and \(x^2 - 4x - 5\) is the denominator. The behavior of the function largely depends on what's happening in the denominator. If the denominator becomes zero, the function doesn't have a valid output, leading to special features in the graph.

Holes in Graphs

Holes in the graph of a rational function occur at values of \(x\) that make both the numerator and denominator zero, which results in an undefined expression that can be simplified away. Essentially, this means a part of the graph is missing, almost like there was a 'hole' in the line.To identify holes, you want to look for any factors that are present in both the numerator and the denominator and then solve for when they are zero. For example:

• With \(f(x) = \frac{x-5}{(x-5)(x+1)}\), notice \(x-5\) is a common factor in both the numerator and denominator.
• Setting \(x-5=0\) gives \(x=5\), which indicates a hole at this point in the graph when \(x-5\) cancels out.

This removal leaves the graph with a subtle indiscernible gap or hole at \(x=5\). Holes are an important concept as they reflect the fine intricacies in the behavior of rational functions.

Factoring Quadratics

Factoring quadratics is a pivotal skill in analyzing rational functions as it allows us to simplify expressions and unravel the secrets hidden in the graphs, like identifying holes or vertical asymptotes.A quadratic is an expression of the form \(ax^2 + bx + c\). To factor it, you look for two numbers that multiply to the constant term \(c\) and add up to the linear coefficient \(b\). Let's take:

• \(x^2 - 4x - 5\)

The goal is to find numbers that multiply to \(-5\) (the constant term) and add up to \(-4\) (the coefficient of \(x\)). Here, the numbers \(-5\) and \(1\) fit perfectly, so the quadratic factors to \((x-5)(x+1)\). This factored form is crucial for identifying points where the function might have a hole or a vertical asymptote, making factoring a valuable method for students dealing with these functions.