Find the Asymptotes f(x)=(1-4x)/(1+7x)
In this problem, the task is to determine the asymptotes of the function \( f(x) = \frac{1-4x}{1+7x} \). Asymptotes are lines to which a graph of a function approaches infinitely closely but typically does not touch or intersect. There are two types of asymptotes that may be considered here: vertical and horizontal (or possibly oblique if the function's degree in the numerator is exactly one higher than the denominator's).
You are asked to analyze the behavior of the function as the independent variable x approaches specific values where the function could go to infinity (which would be vertical asymptotes) and as x approaches positive or negative infinity (which would suggest horizontal or oblique asymptotes). This typically involves looking at the limits of the function and examining the behavior of the function at values of x that make the denominator zero (for vertical asymptotes) or at extreme values of x for horizontal or oblique asymptotes.
$f \left(\right. x \left.\right) = \frac{1 - 4 x}{1 + 7 x}$
Answer
Solution:
Step 1:
Identify the values of $x$ that cause the function $\frac{1 - 4x}{1 + 7x}$ to be undefined. This occurs when the denominator equals zero: $x = -\frac{1}{7}$.
Step 2:
Examine the general form of 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. 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 3:
Determine the values of $n$ and $m$ for our function. Here, $n = 1$ and $m = 1$.
Step 4:
Since $n$ equals $m$, we find the horizontal asymptote by dividing the leading coefficients: $y = \frac{a}{b}$, with $a = -4$ and $b = 7$. Thus, $y = -\frac{4}{7}$.
Step 5:
An oblique asymptote is not present as 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 Asymptotes: $x = -\frac{1}{7}$
- Horizontal Asymptotes: $y = -\frac{4}{7}$
- No Oblique Asymptotes
Step 7:
The process is complete.
Knowledge Notes:
Asymptotes are lines that a graph of a function approaches but never touches. There are three types of asymptotes: vertical, horizontal, and oblique (also called slant).
Vertical Asymptotes: These occur at values of $x$ where the function becomes undefined, typically where the denominator of a rational function is zero. To find them, solve the equation where the denominator equals zero.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator ($n$) and the denominator ($m$) in a rational function. 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, respectively. If $n > m$, 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. They are found by performing polynomial long division or synthetic division.
Rational Functions: A rational function is a ratio of two polynomials, written in the form $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.
Undefined Points: A function is undefined at points where it would require division by zero, which is not possible in standard arithmetic.
In the given problem, the function $f(x) = \frac{1 - 4x}{1 + 7x}$ has a vertical asymptote where the denominator is zero, and a horizontal asymptote determined by the ratio of the leading coefficients since the degrees of the numerator and denominator are equal. There are no oblique asymptotes since the degree of the numerator is not greater than the degree of the denominator.
