Find the Asymptotes f(x)=(7x)/(x^2-6x)
The problem involves finding the asymptotes of the function f(x) = (7x)/(x^2-6x). Asymptotes are lines that the graph of a function approaches but never actually reaches as the value of x moves towards infinity or a specific value causing a discontinuity. There are two types of asymptotes typically considered in such problems: vertical and horizontal (or oblique, for functions that do not have horizontal asymptotes).
Vertical asymptotes occur where the function is undefined, which usually corresponds to values of x that make the denominator zero.
Horizontal or oblique asymptotes involve the behavior of the function as x approaches infinity or negative infinity, indicating the function's end behavior.
The question would likely require you to identify both types of asymptotes for the given rational function if they exist.
$f \left(\right. x \left.\right) = \frac{7 x}{x^{2} - 6 x}$
Answer
Solution:
Step 1:
Identify the values for which the function $\frac{7x}{x^2 - 6x}$ does not exist. These are $x = 0$ and $x = 6$.
Step 2:
Observe the behavior of the function as $x$ approaches $6$. The function tends to negative infinity when approaching $6$ from the left and to positive infinity when approaching from the right. Hence, $x = 6$ is a vertical asymptote.
Step 3:
For a general rational function $R(x) = \frac{ax^n}{bx^m}$, 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 is no horizontal asymptote, but there may be an oblique asymptote.
Step 4:
Determine the values of $n$ and $m$. For our function, $n = 1$ and $m = 2$.
Step 5:
Given that $n < m$, the horizontal asymptote is the x-axis, which is $y = 0$.
Step 6:
An oblique asymptote is not present since the degree of the numerator ($n$) is not greater than the degree of the denominator ($m$).
Step 7:
Compile the list of asymptotes for the function:
- Vertical Asymptote: $x = 6$
- Horizontal Asymptote: $y = 0$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, one must understand the different types of asymptotes and how they relate to the function's behavior at various points.
Vertical Asymptotes: These occur at values of $x$ where the function is undefined and typically result from setting the denominator equal to zero. The function will approach infinity or negative infinity near these points.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and the denominator ($n$ and $m$ respectively). If the degree of the numerator is less than the degree of the denominator ($n < m$), the horizontal asymptote is $y = 0$. If the degrees are equal ($n = m$), the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively. If the numerator's degree is greater ($n > m$), there is no horizontal asymptote.
Oblique Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator ($n = m + 1$). In such cases, one must perform long division of the numerator by the denominator to find the equation of the oblique asymptote.
Undefined Points: It is important to identify where the function is undefined, as these points often lead to vertical asymptotes.
Behavior at Infinity: Understanding the behavior of the function as $x$ approaches infinity or negative infinity can help determine the presence of horizontal or oblique asymptotes.
In the given problem, the function $\frac{7x}{x^2 - 6x}$ has a vertical asymptote at $x = 6$ and a horizontal asymptote at $y = 0$. There is no oblique asymptote because the degree of the numerator is less than the degree of the denominator.
