Find the Asymptotes f(x)=(x^2+x-6)/(x^2-4x-21)
In the given problem, you are asked to determine the asymptotes of the function f(x) = (x^2 + x - 6) / (x^2 - 4x - 21). Asymptotes are lines that the graph of the function approaches as x goes to infinity or minus infinity. There are two types of asymptotes that you may need to find: vertical and horizontal (or oblique).
Vertical asymptotes occur when the denominator of the function approaches zero, which could cause the function value to approach infinity. For rational functions, vertical asymptotes are found by setting the denominator equal to zero and solving for x, ensuring that these solutions are not also zeros of the numerator (since such a point would be a hole rather than an asymptote).
Horizontal asymptotes are determined by the end behavior of the function as x approaches infinity or negative infinity. They can be found by comparing the degrees of the polynomials in the numerator and denominator or by simplifying the function's expression as x approaches large magnitudes.
Your question is to identify these lines for the given function by examining its algebraic structure and the limits of the function as x approaches certain critical values.
$f \left(\right. x \left.\right) = \frac{x^{2} + x - 6}{x^{2} - 4 x - 21}$
Answer
Solution:
Step 1:
Determine the values of $x$ that make the function $\frac{x^2 + x - 6}{x^2 - 4x - 21}$ undefined. These are $x = -3$ and $x = 7$.
Step 2:
Observe the behavior of $\frac{x^2 + x - 6}{x^2 - 4x - 21}$ as $x$ approaches $7$. It tends towards $-\infty$ from the left and $\infty$ from the right, indicating $x = 7$ is a vertical asymptote.
Step 3:
For a rational function $R(x) = \frac{a x^n}{b x^m}$, where $n$ is the degree of the numerator and $m$ is the degree of the denominator:
If $n < m$, the horizontal asymptote is $y = 0$.
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$.
If $n > m$, there are no horizontal asymptotes, but there may be an oblique asymptote.
Step 4:
Identify the degrees $n$ and $m$ of the numerator and denominator, respectively. Here, $n = 2$ and $m = 2$.
Step 5:
Since the degrees $n$ and $m$ are equal, the horizontal asymptote is given by $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively. In this case, $y = 1$.
Step 6:
There are no oblique asymptotes since the degree of the numerator is not greater than the degree of the denominator.
Step 7:
Compile the list of all asymptotes for the function:
- Vertical Asymptotes: $x = 7$
- Horizontal Asymptotes: $y = 1$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, follow these steps:
Vertical Asymptotes: These occur at the values of $x$ that make the denominator $Q(x)$ equal to zero, provided that the numerator $P(x)$ does not also equal zero at these points. To find them, solve $Q(x) = 0$.
Horizontal Asymptotes: These depend on the degrees of the numerator and denominator polynomials.
If the degree of $P(x)$ is less than the degree of $Q(x)$, the horizontal asymptote is $y = 0$.
If the degrees are equal, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of $P(x)$ and $Q(x)$, respectively.
If the degree of $P(x)$ is greater than the degree of $Q(x)$, there is no horizontal asymptote.
Oblique Asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator, the function may have an oblique asymptote. To find it, perform polynomial long division of $P(x)$ by $Q(x)$.
Behavior Around Vertical Asymptotes: To determine how the function behaves around its vertical asymptotes, analyze the limits of $f(x)$ as $x$ approaches the vertical asymptote from the left and right.
Combining Asymptotes: The set of all asymptotes provides a sketch of the function's behavior at infinity and near critical points where the function is undefined.
