Find the Asymptotes f(x)=(1-6x)/(1+7x)
The given problem is asking to determine the lines that the function f(x) = (1-6x)/(1+7x) approaches as its independent variable, x, goes to infinity or negative infinity, or where the function becomes undefined. These lines are known as asymptotes. There can be vertical and horizontal (or potentially oblique) asymptotes, and the goal is to calculate their equations if they exist. The vertical asymptotes are typically found where the denominator of the function is zero (since division by zero is undefined), while horizontal asymptotes are determined by analyzing the behavior of the function as x approaches positive or negative infinity. Oblique asymptotes may occur when the degree of the numerator is exactly one higher than the degree of the denominator, and they are formed by the slant or tilted line that the graph of the function approaches. The task involves algebraic manipulation and limits to find these characteristics of the function.
$f \left(\right. x \left.\right) = \frac{1 - 6 x}{1 + 7 x}$
Answer
Solution:
Step 1:
Determine the points at which the function $\frac{1 - 6x}{1 + 7x}$ does not exist. This occurs when $x = -\frac{1}{7}$.
Step 2:
Examine the general form of a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ and $m$ are the degrees of the polynomial in the numerator and denominator, respectively.
A horizontal asymptote at $y = 0$ exists if $n < m$.
If $n = m$, the horizontal asymptote is found at $y = \frac{a}{b}$.
When $n > m$, there is no horizontal asymptote, but there may be an oblique asymptote.
Step 3:
Identify the values of $n$ and $m$ for the given function. We find $n = 1$ and $m = 1$.
Step 4:
Since $n$ equals $m$, we have a horizontal asymptote at $y = \frac{a}{b}$, with $a = -6$ and $b = 7$. Thus, the horizontal asymptote is $y = -\frac{6}{7}$.
Step 5:
An oblique asymptote is not present since the degree of the numerator is not greater than the degree of the denominator.
Step 6:
Compile the list of asymptotes for the function:
- Vertical Asymptote: $x = -\frac{1}{7}$
- Horizontal Asymptote: $y = -\frac{6}{7}$
- No Oblique Asymptotes
Knowledge Notes:
To solve for the asymptotes of a rational function, one must understand the following key concepts:
Undefined Points: A rational function is undefined where its denominator equals zero. These points are potential vertical asymptotes.
Degrees of Polynomials: The degree of a polynomial is the highest power of $x$ in its expression. The degrees of the numerator and denominator polynomials are denoted by $n$ and $m$, respectively.
Horizontal Asymptotes: These occur depending on the relationship between $n$ and $m$:
If $n < m$, the horizontal asymptote is the x-axis ($y = 0$).
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, 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 ($n = m + 1$), the function may have an oblique (or slant) asymptote, which can be found using polynomial long division or synthetic division.
Vertical Asymptotes: These are found by setting the denominator equal to zero and solving for $x$, provided that the numerator does not also equal zero at those points (which would indicate a hole in the graph rather than an asymptote).
By applying these concepts to the given function, one can determine the presence and equations of any vertical, horizontal, or oblique asymptotes.
