Find the Asymptotes f(x)=(2-x-x^3)/(x^3-1)
The problem is asking for the determination of the asymptotes of the function f(x)=(2-x-x^3)/(x^3-1). An asymptote is a line that the graph of the function approaches but never actually reaches as x approaches infinity or minus infinity, or at certain points where the function is undefined due to a discontinuity (like a hole or vertical asymptote). The task involves analyzing the behavior of the function at these points and as x tends toward positive or negative infinity in order to identify any horizontal, vertical, or slant (oblique) asymptotes. Typically, this includes simplifying the function if possible, and applying limits to investigate the behavior of the function at its critical points and at infinity.
$f \left(\right. x \left.\right) = \frac{2 - x - x^{3}}{x^{3} - 1}$
Answer
Solution:
Step:1 Determine the values of $x$ for which the function $\frac{2 - x - x^{3}}{x^{3} - 1}$ is not defined. This occurs when the denominator equals zero. Thus, $x = 1$.
Step:2 Identify the vertical asymptotes, which are typically located where the function is not defined and the limit approaches infinity. In this case, there are no vertical asymptotes.
Step:3 For a rational function $R(x) = \frac{a x^{n}}{b x^{m}}$, with $n$ as the highest power in the numerator and $m$ as the highest power in the denominator, follow these rules:
If $n < m$, the horizontal asymptote is $y = 0$.
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$.
If $n > m$, there are no horizontal asymptotes; instead, look for an oblique asymptote.
Step:4 Calculate the degrees $n$ and $m$ of the numerator and denominator, respectively. For our function, $n = 3$ and $m = 3$.
Step:5 Given that $n = m$, the horizontal asymptote can be found using $y = \frac{a}{b}$. The leading coefficients are $a = -1$ and $b = 1$, hence $y = -1$.
Step:6 An oblique asymptote is not present since the degree of the numerator is not greater than the degree of the denominator. Therefore, we do not have any oblique asymptotes.
Step:7 Compile the complete list of asymptotes for the function:
- No Vertical Asymptotes
- Horizontal Asymptote: $y = -1$
- No Oblique Asymptotes
Knowledge Notes:
Rational Functions: A rational function is a function that can be expressed as the quotient of two polynomials. It is of the form $R(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.
Asymptotes: Asymptotes are lines that the graph of a function approaches as $x$ or $y$ goes to infinity. There are three types of asymptotes: vertical, horizontal, and oblique (or slant).
Vertical Asymptotes: These occur at values of $x$ where the function goes to infinity. They are found by setting the denominator equal to zero and solving for $x$.
Horizontal Asymptotes: These are horizontal lines that the graph approaches as $x$ goes to infinity. For a rational function $R(x) = \frac{a x^{n}}{b x^{m}}$, if $n < m$, the horizontal asymptote is $y = 0$; if $n = m$, it's $y = \frac{a}{b}$; and if $n > m$, there are no horizontal asymptotes.
Oblique Asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator, the graph may have an oblique asymptote. This is found by performing polynomial long division or synthetic division.
Undefined Points: A function is undefined at points where the denominator is zero. These points are not part of the function's domain.
Leading Coefficient: The leading coefficient of a polynomial is the coefficient of the term with the highest power.
Understanding these concepts is crucial for analyzing the behavior of rational functions and determining their asymptotes.
