Problem 23 - College Algebra

Question

Find the vertical asymptotes, if any, of the graph of each rational function. $$g(x)=\frac{x+3}{x(x+4)}$$

Step-by-Step Solution

Answer

The vertical asymptotes of the given rational function \(g(x)=\frac{x+3}{x(x+4)}\) are at \(x = 0\) and \(x = -4\).

Step 1: Identify the Rational Function

The given rational function is \(g(x)=\frac{x+3}{x(x+4)}\)

Step 2: Set the Denominator Equal to Zero

To find the vertical asymptotes, set the denominator of the rational function equal to zero and solve for \(x\). That is \(x(x+4) = 0\)

Step 3: Solve for x

If \(x(x+4) = 0\), either \(x = 0\) or \(x = -4\)

Step 4: Checking restrictions

We already established that \(x = 0\) and \(x = -4\) makes the denominator zero. Hence, \(x = 0\) and \(x = -4\) are the vertical asymptotes. However, let's verify if the numerator becomes zero at these values. Substituting \(x = 0\) in the numerator, it results to 3 and for \(x = -4\) it gives -1. So, there are indeed vertical asymptotes at these two points.

Key Concepts

Rational Functions

Rational functions are a special type of mathematical expression. They are expressed as the ratio of two polynomials. In simpler terms, a rational function is written using fractions where both the numerator and the denominator are polynomials. A basic example of a rational function is given by \(g(x) = \frac{x+3}{x(x+4)}\). Here, the numerator is \(x+3\) and the denominator is \(x(x+4)\).

• Form: \(\frac{P(x)}{Q(x)}\) where both \(P(x)\) and \(Q(x)\) are polynomial functions.
• Polynomials are sums of terms each consisting of a variable raised to a power and multiplied by a coefficient.

Understanding rational functions is key to graphing them and analyzing their behavior, especially at points where the functions become undefined.

Denominator

The denominator of a rational function plays a critical role in determining the function's behavior. It is the part of the expression located below the fraction line. In the function \(g(x)=\frac{x+3}{x(x+4)}\), the denominator is \(x(x+4)\).

• Importance: Determines where the rational function is undefined by setting it equal to zero.
• Calculation: Solving \(x(x+4) = 0\) reveals the values that lead to division by zero, indicating possible vertical asymptotes.

When dealing with rational functions, it's crucial to understand how the denominator affects the graph and makes determining its zero crucial in finding vertical asymptotes.

Numerator

The numerator of a rational function is the expression above the division line. In \(g(x) = \frac{x+3}{x(x+4)}\), the numerator is \(x+3\). It influences the values of the function but doesn't affect the undefined points unless it also becomes zero at the points where the denominator is zero.

• If the numerator is zero, the function value at that point is zero unless the denominator is also zero.
• Critical in further determining behaviors on the graph beyond just vertical asymptotes.

For vertical asymptotes, once zeros from the denominator are checked, it's also essential to verify that the numerator is non-zero at those points for true asymptotes to exist.

Asymptotes in Graphs

Asymptotes, particularly vertical asymptotes, are lines that the graph of a function approaches but never touches or crosses. They occur in rational functions where the denominator is zero. For instance, in \(g(x) = \frac{x+3}{x(x+4)}\), we found vertical asymptotes at \(x = 0\) and \(x = -4\).

• Characterize places where the function grows indefinitely positive or negative.
• Vertical asymptotes are solutions to the equation formed by setting the denominator equal to zero.

As part of graph analysis, identifying asymptotes helps in predicting and sketching the general shape and behavior of the function across its domain.