Find the Asymptotes f(x)=(x^2-8x+15)/(x^2-7x+12)
The problem asks for the identification of the asymptotes of the function \( f(x) = \frac{x^2-8x+15}{x^2-7x+12} \). An asymptote is a line that the graph of a function approaches but never actually reaches. The two types of asymptotes that are relevant for rational functions like this one are horizontal asymptotes, which occur as the input, x, approaches infinity or negative infinity, and vertical asymptotes, which occur at values of x for which the function is undefined due to the denominator being zero. This problem involves analyzing the function's behavior at infinity for horizontal asymptotes and finding the roots of the denominator for potential vertical asymptotes.
$f \left(\right. x \left.\right) = \frac{x^{2} - 8 x + 15}{x^{2} - 7 x + 12}$
Answer
Solution:
Step 1:
Determine the values for which the function $\frac{x^{2} - 8x + 15}{x^{2} - 7x + 12}$ does not exist. These are $x = 3$ and $x = 4$.
Step 2:
Observe that as $x$ approaches $4$ from the left, $\frac{x^{2} - 8x + 15}{x^{2} - 7x + 12}$ tends towards infinity, and from the right, it tends towards negative infinity. Thus, $x = 4$ is a vertical asymptote.
Step 3:
Examine the general form of a rational function $R(x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the highest power in the numerator and $m$ is the highest power in the denominator. The rules for horizontal asymptotes are as follows:
If $n < m$, the horizontal asymptote is $y = 0$.
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$.
If $n > m$, there is no horizontal asymptote, but there may be an oblique asymptote.
Step 4:
Identify the degrees $n$ and $m$ for the numerator and denominator, respectively. In this case, $n = 2$ and $m = 2$.
Step 5:
Since the degrees of the numerator and denominator are equal ($n = m$), the horizontal asymptote can be found by dividing the leading coefficients. Here, $a = 1$ and $b = 1$, so the horizontal asymptote is $y = \frac{1}{1}$ or $y = 1$.
Step 6:
Conclude that 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 asymptotes for the function:
- Vertical Asymptotes: $x = 4$
- Horizontal Asymptotes: $y = 1$
- No Oblique Asymptotes
Step 8:
(No action required for this step as it is blank in the original process.)
Knowledge Notes:
When analyzing rational functions for asymptotes, there are three types to consider: vertical, horizontal, and oblique.
Vertical Asymptotes: These occur at values of $x$ that make the denominator zero (and the numerator non-zero), as the function approaches infinity or negative infinity at these points.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and the denominator ($n$ and $m$ respectively). If $n < m$, the horizontal asymptote is $y = 0$. If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator. If $n > m$, there is no horizontal asymptote.
Oblique Asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator ($n = m + 1$), the function may have an oblique asymptote, which is found using polynomial long division.
In the given problem, we have a rational function where the degrees of the numerator and denominator are equal, leading to a horizontal asymptote at $y = \frac{a}{b}$, and since there are values that make the denominator zero, we also have vertical asymptotes at those points. There are no oblique asymptotes as the degree of the numerator is not greater than the degree of the denominator.
