Problem 25 - College Algebra

Question

Find the vertical asymptotes, if any, of the graph of each rational function. $$h(x)=\frac{x}{x(x+4)}$$

Step-by-Step Solution

Answer

The function \(h(x)=\frac{x}{x(x+4)}\) has one vertical asymptote at \(x=-4\).

Step 1: Simplify the function

First, simplify the function \(h(x)=\frac{x}{x(x+4)}\). The \(x\) in the numerator and one of the \(x\) in the denominator cancel, giving \(h(x)=\frac{1}{x+4}\).

Step 2: Find where the denominator equals zero

Second, set the denominator equal to zero to find potential vertical asymptotes. \(x+4=0\) so \(x=-4\).

Step 3: Check the numerator

Check the numerator of the original function at \(x=-4\). \(x=-4\) does not make the numerator zero, so it is indeed a vertical asymptote.

Key Concepts

Rational Functions

Rational functions are mathematical expressions that represent the ratio of two polynomials. They are highly important in calculus and algebra, as they can model a wide variety of real-world phenomena.

A rational function is typically expressed as:

• \( R(x) = \frac{P(x)}{Q(x)} \)

Where \( P(x) \) and \( Q(x) \) are polynomial functions and \( Q(x) eq 0 \) (because you cannot divide by zero).

These rational expressions can have unique features such as asymptotes, which are lines that the graph of the function approaches but never crosses. Identifying these features is vital for understanding the behavior of the function, especially when analyzing limits and continuity.

Simplifying Rational Expressions

Simplifying rational expressions is a key step to make them easier to work with and to reveal important characteristics such as asymptotes. It involves reducing the expression to its simplest form by canceling out common factors in the numerator and the denominator.

In the original exercise, the rational expression \( h(x) = \frac{x}{x(x+4)} \) was simplified by canceling out the common factor of \( x \) from both the numerator and the denominator, leading to \( h(x) = \frac{1}{x+4} \).

Steps to simplify a rational expression include:

• Factor both the numerator and the denominator completely.
• Cancel out any common factors.

Simplifying helps in easily identifying potential vertical asymptotes or holes in the graph of the function.

Finding Asymptotes

Finding asymptotes, especially vertical ones, is crucial in the analysis of rational functions to understand their limits and continuity.

Vertical asymptotes occur where the denominator of a rational function is equal to zero, as the function tends toward infinity. This is evident in the original exercise, where we explored \( h(x)=\frac{1}{x+4} \).

To find vertical asymptotes:

• Set the simplified denominator equal to zero.
• Solve for \( x \). Any value of \( x \) that satisfies this equation indicates a vertical asymptote unless the same value makes the numerator zero in the original non-simplified form. In this case, it may be a hole instead.

In the example provided, setting \( x+4=0 \) results in \( x=-4 \), identifying a vertical asymptote, since it doesn't cancel with a factor in the numerator.