Problem 100 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find any asymptotes for the graph of $$ R(x)=\frac{x+3}{x^{2}-x-12} $$

Step-by-Step Solution

Answer

Vertical asymptote at \(x = 4\), horizontal asymptote at \(y = 0\), and a removable discontinuity at \(x = -3\).

Step 1: Factorize the denominator

The function given is \[R(x) = \frac{x+3}{x^2 - x - 12}\]. The first step is to factorize the denominator. We need to find two numbers that multiply to give -12 and add to give -1. The factors are -4 and 3. So, \[ x^2 - x - 12 = (x - 4)(x + 3)\].

Step 2: Check for any common factors

Checking the numerator and the factorized denominator for any common factors. Since the numerator is \(x + 3\) and one of the denominator's factors is also \(x + 3\), we can cancel these out. Hence, the simplified function becomes \[R(x) = \frac{1}{x - 4}\] with the restriction that \(x eq -3\).

Step 3: Identify vertical asymptotes

Vertical asymptotes occur where the denominator is zero (and the numerator is not zero). For the simplified function \(R(x) = \frac{1}{x - 4}\), the denominator becomes zero when \(x = 4\). Hence, there is a vertical asymptote at \(x = 4\). Additionally, there is an original denominator zero at \(x = -3\) in the unreduced form, which doesn't cancel out in \(R(x)\), indicating a removable discontinuity or hole at \(x = -3\).

Step 4: Identify horizontal asymptotes

Horizontal asymptotes are determined by comparing the degrees of the polynomials in the numerator and the denominator. The original function has a numerator of degree 1 and a denominator of degree 2. When the degree of the denominator is higher, \(y = 0\) is a horizontal asymptote. Therefore, the horizontal asymptote is \(y = 0\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function is zero and the numerator is not zero at that point. To find vertical asymptotes, you need to determine the values of x that make the denominator equal to zero.
In the given function \[ R(x) = \frac{x+3}{x^2 - x - 12} \], we factorized the denominator as \[ (x - 4)(x + 3) \].
After simplifying, the function becomes \[ R(x) = \frac{1}{x - 4} \]. For this simplified function, the denominator becomes zero at \(x = 4\), indicating a vertical asymptote at \(x = 4\).
Additionally, the original form of the function had a zero at \(x = -3\). Since it cancels out, it represents a removable discontinuity rather than a vertical asymptote.

Horizontal Asymptotes

Horizontal asymptotes are lines that the graph of a function approaches as \(x\) goes to infinity or negative infinity. To find horizontal asymptotes for rational functions, compare the degrees of the numerator and denominator polynomials.
For the function \[ R(x) = \frac{x+3}{x^2 - x - 12} \], the degree of the numerator is 1 (since the highest power of x is \(x^1\)) and the degree of the denominator is 2 (since the highest power of x is \(x^2\)).
When the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is \(y = 0\). Thus, \(y = 0\) is the horizontal asymptote for this function.

Rational Function Simplification

Simplifying rational functions involves reducing the function to its simplest form by canceling common factors in the numerator and the denominator.
For \[ R(x) = \frac{x+3}{x^2 - x - 12} \], we factorize the denominator as \[ (x - 4)(x + 3) \]. Then, by canceling the common factor \( (x + 3) \) from the numerator and denominator, we simplify the function to \[ R(x) = \frac{1}{x - 4} \].
Remember, the simplification excludes values that would originally make the denominator zero. In this case, \(x eq -3\) as it cancels out, indicating a hole in the graph at that point.

Factorization in Algebra

Factorization is a technique used to break down polynomials into products of simpler polynomials. It's crucial for simplifying rational functions and finding roots.
For the equation \[ x^2 - x - 12 \], we need to find two factors that multiply to \(-12\) (the constant term) and add up to \(-1\) (the coefficient of x). The factors are \(-4\) and \(3\). Thus, the factorized form of \[ x^2 - x - 12 \] is \[ (x - 4)(x + 3) \].
Using factorization helps in identifying vertical asymptotes and simplifying rational functions, as seen in our example where we reduce \[ R(x) = \frac{x+3}{x^2 - x - 12} \] to \[ R(x) = \frac{1}{x - 4} \].