Find the Asymptotes f(x)=(3x^2-13x+4)/(x^2-3x-4)
The problem provided is a question from calculus and precalculus, which involves finding the asymptotes of a given rational function. An asymptote is a line that the graph of the function approaches but never actually reaches. In this particular problem, the function $f(x)=(3x^2-13x+4)/(x^2-3x-4)$is a rational function, where both the numerator and the denominator are polynomials.
The question requires one to:
Identify and find any vertical asymptotes, which occur where the denominator of the rational function is equal to zero, unless the numerator also equals zero at the same points (indicating a possible hole in the graph rather than an asymptote).
Determine any horizontal asymptotes by examining the degrees of the polynomial in the numerator and the denominator and applying the relevant rules depending on whether the degree of the numerator is less than, equal to, or greater than the degree of the denominator.
Optionally, if applicable, calculate any oblique (slant) asymptotes, which might occur when the degree of the numerator is exactly one higher than the degree of the denominator. This involves performing polynomial division to find the equation of the slant asymptote.
The problem does not require solving or simplifying the function but rather finding these specific lines that the graph of the function approaches as $x$approaches infinity or negative infinity, or where the function is undefined.
$f \left(\right. x \left.\right) = \frac{3 x^{2} - 13 x + 4}{x^{2} - 3 x - 4}$
Answer
Solution:
Step 1:
Determine the values for which the function $\frac{3x^2 - 13x + 4}{x^2 - 3x - 4}$ does not exist. These values are $x = -1$ and $x = 4$.
Step 2:
Examine the behavior of the function as $x$ approaches $-1$. The function tends to positive infinity when approaching from the left and to negative infinity from the right. Thus, $x = -1$ is a vertical asymptote.
Step 3:
To find horizontal asymptotes, consider the highest power terms in the numerator and denominator of $R(x) = \frac{ax^n}{bx^m}$. The rules 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 of the numerator and denominator. Here, $n = 2$ and $m = 2$.
Step 5:
Since the degrees are equal ($n = m$), the horizontal asymptote is determined by the ratio of the leading coefficients $a$ and $b$. With $a = 3$ and $b = 1$, the horizontal asymptote is $y = 3$.
Step 6:
An oblique asymptote does not exist as the degree of the numerator is not greater than the degree of the denominator.
Step 7:
The complete set of asymptotes is given by:
- Vertical Asymptote: $x = -1$
- Horizontal Asymptote: $y = 3$
- No Oblique Asymptotes
Knowledge Notes:
To solve for the asymptotes of a rational function, one must understand the following concepts:
Vertical Asymptotes: These occur at values of $x$ where the denominator of a rational function is zero and the numerator is not zero. The function approaches infinity or negative infinity as $x$ approaches these values.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator ($n$) and denominator ($m$) of the rational function in its simplified form. If $n < m$, the horizontal asymptote is the x-axis ($y = 0$). If $n = m$, the horizontal asymptote is the line $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively. 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, the rational function may have an oblique asymptote. This is found by performing polynomial long division or synthetic division to find the slant asymptote equation.
Undefined Values: A rational function is undefined where its denominator is zero. These points are potential locations for vertical asymptotes.
Behavior at Asymptotes: To determine the exact nature of a vertical asymptote, one must analyze the limit of the function as it approaches the asymptote from both the left and the right.
By applying these principles, one can determine all vertical, horizontal, and oblique asymptotes for a given rational function.
