Problem 24 - Algebra and Trigonometry

Question

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each crational function. $$ g(x)=\frac{x+3}{x(x-3)} $$

Step-by-Step Solution

Answer

The vertical asymptotes of the function are \( x = 0 \) and \( x = 3 \). The function has a hole at \( x = -3 \).

Step 1: Find the Vertical Asymptotes

To find the vertical asymptotes, set the denominator of the rational function equal to zero and solve for \( x \). \n\nSo,\n\n \(x(x-3)=0\)\n\nSolution of this equation will give the vertical asymptotes.

Step 2: Find the Holes

To find the holes, set both the numerator and denominator of the rational function equal to zero and solve for \( x \).\n\nSo,\n\n \(x+3=0\) and \(x(x-3)=0\)\n\nIf there is any common solution, it will correspond to a hole.

Step 3: Solve the Equations

After solving these equations in steps 1 and 2 , we get \( x = 0 \), \( x = 3 \), and \( x = -3 \). \n\nAs \( x = 0 \) and \( x = 3 \) are only solutions of denominator so, these are vertical asymptotes. \n\nAnd \( x = -3 \) is the solution for both numerator and denominator, so it's the hole of the given function.

Key Concepts

Rational Functions

A rational function is a type of function represented as the ratio of two polynomial expressions. The general form is:

• The numerator is the top polynomial, which determines the output value of the function.
• The denominator is the bottom polynomial, crucial in identifying potential points of undefined values in the function.

For an example, consider the function provided in the exercise, which is given by:\[ g(x) = \frac{x+3}{x(x-3)} \]Here, the numerator is \(x+3\), and the denominator is \(x(x-3)\). These elements together allow us to analyze various characteristics of the function, including vertical asymptotes and holes, which indicate specific behaviors and inputs that cause the function to be undefined.

Holes in Graphs

In the context of rational functions, a hole is a point on the graph where the function is not defined, even though there is no vertical asymptote at that location. This typically occurs when there is a common factor in both the numerator and the denominator of the rational function.
To find a hole in the graph of a rational function, follow these steps:

• Set both the numerator and the denominator to zero.
• Solve these equations to find any common solutions.

If a solution satisfies both the numerator and the denominator being zero, it indicates a hole. For the function \( g(x) = \frac{x+3}{x(x-3)} \), setting \(x+3=0\) gives \(x=-3\), and setting \(x(x-3)=0\) gives \(x=0\) and \(x=3\). Since \(x=-3\) is a solution for both, it represents a hole, unlike the vertical asymptotes which are points where the function grows infinitely.

Denominator Zero

When examining rational functions, one critical value is when the denominator equals zero, as it indicates points where the function becomes undefined and may suggest vertical asymptotes.
Here's what to do:

• Set the denominator of the function equal to zero.
• Solve the resulting equation for \(x\) to identify possible vertical asymptotes.

In the given exercise, the denominator is \(x(x-3)\). By setting it to zero:\[ x(x-3) = 0 \]The solutions \(x = 0\) and \(x = 3\) are found, indicating these points could be vertical asymptotes. Points where the denominator is zero but the numerator is not zero generally indicate vertical asymptotes, marking x-values where the function skyrockets toward positive or negative infinity, and not holes, which occur at specific conditions of common factors with the numerator.