Problem 21 - APEX Calculus

Question

Identify the horizontal and vertical asymptotes, if any, of the given function. $$ f(x)=\frac{x^{2}+x-12}{7 x^{3}-14 x^{2}-21 x} $$

Step-by-Step Solution

Answer

Vertical asymptotes: \( x = 0 \), \( x = -1 \); Horizontal asymptote: \( y = 0 \).

Step 1: Finding Vertical Asymptotes

Vertical asymptotes occur where the function is undefined, which typically happens when the denominator equals zero. Set the denominator equal to zero and solve for \(x\):\[ 7x^3 - 14x^2 - 21x = 0 \]Factor out the greatest common factor, \(7x\):\[ 7x(x^2 - 2x - 3) = 0 \]Set each factor equal to zero:1. \( 7x = 0 \) gives \( x = 0 \).2. Solve \( x^2 - 2x - 3 = 0 \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 1 \), \( b = -2 \), \( c = -3 \): \[ x = \frac{2 \pm \sqrt{4 + 12}}{2} = \frac{2 \pm \sqrt{16}}{2} = \frac{2 \pm 4}{2} \] This gives \( x = 3 \) and \( x = -1 \).The vertical asymptotes are \( x = 0 \), \( x = 3 \), and \( x = -1 \).

Step 2: Checking for Common Factors

Before finding horizontal asymptotes, check for any common factors between the numerator and denominator that might cancel out, which affects vertical asymptotes. Factor the numerator:\[ x^2 + x - 12 = (x - 3)(x + 4) \]Now, reconsider the denominator:\[ 7x(x^2 - 2x - 3) = 7x(x - 3)(x + 1) \]The common factor \((x-3)\) cancels out. The reduced function is:\[ f(x) = \frac{(x + 4)}{7x(x + 1)} \]Now, there are vertical asymptotes at \( x = 0 \) and \( x = -1 \). \( x = 3 \) is a removable discontinuity.

Step 3: Finding Horizontal Asymptotes

To find the horizontal asymptotes, consider the degrees of the numerator and the denominator.The degree of the numerator is 2, and the degree of the denominator is 3. Since the degree of the denominator is greater than the numerator, the horizontal asymptote is \( y = 0 \).

Key Concepts

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that the graph of a function approaches as the input, or x-value, extends to positive or negative infinity. Horizontal asymptotes help predict the end behavior of a function and occur in rational functions based on the relationship between the degrees of the numerator and the denominator.

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the x-axis:

• This situation is common when rational functions have higher degree denominators.
• In this case, for the function \( f(x) = \frac{x^2 + x - 12}{7x^3 - 14x^2 - 21x} \), the numerator's degree (2) is less than the denominator's degree (3), resulting in a horizontal asymptote at \( y = 0 \).

Thus, for large values of x, the function gets closer to this line but never actually touches it.

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the function is undefined due to division by zero in the denominator. They appear as vertical lines on the graph.

To find vertical asymptotes:

• Set the denominator equal to zero and solve for x. This helps identify where the function goes to positive or negative infinity.
• For our example function \( f(x) = \frac{x^2 + x - 12}{7x^3 - 14x^2 - 21x} \), initially setting the denominator \( 7x^3 - 14x^2 - 21x = 0 \) gives multiple roots. By factoring, \( x = 0 \), \( x = 3 \), and \( x = -1 \) are solutions.
• Common factors between the numerator and denominator can cancel out some potential vertical asymptotes. In this case, the factor \( (x-3) \) reduces, leaving \( x = 0 \) and \( x = -1 \) as vertical asymptotes, while \( x = 3 \) becomes a removable discontinuity.

It is crucial to consider these factors to accurately define the vertical asymptotes.

Rational Functions

Rational functions are quotients of polynomials, characterized by having a numerator and a denominator. These functions are intriguing due to their complex behavior and interesting asymptotic features.

Key properties include:

• Rational functions may have asymptotes, which occur where the function approaches infinity or a constant as x approaches specific values or extends to infinity.
• The degree of the numerator and denominator affects their asymptotes. The examples \( f(x) = \frac{x^2 + x - 12}{7x^3 - 14x^2 - 21x} \) demonstrate this, with varying degrees impacting horizontal asymptotes while factors and roots determine vertical characteristics.
• Understanding rational functions includes recognizing how simplifying functions by canceling common factors may change the apparent nature of the function itself, thus affecting the graph and its continuity.

Exploring rational functions deepens mathematical comprehension, emphasizing problem-solving skills in factoring and understanding function behavior at infinite or indeterminate points.