Find the Asymptotes (t-9)/(t^2+81)
The question is asking to determine the asymptotes of the rational function f(t) = (t-9)/(t²+81). An asymptote is a line that the graph of a function approaches but never actually reaches. There are typically two types of asymptotes in the context of rational functions: vertical and horizontal (or oblique/slant in some cases).
Vertical asymptotes occur where the denominator of a fraction goes to zero (unless the numerator also goes to zero at the same point, potentially cancelling the indeterminate form).
Horizontal or slant asymptotes relate to the behavior of the graph as the variable goes to positive or negative infinity.
To solve for these asymptotes, one would typically factor the numerator and denominator (if possible), simplify the expression if common factors exist, and then determine the values which make the denominator zero for vertical asymptotes and analyze the end behavior of the function for horizontal or slant asymptotes.
$\frac{t - 9}{t^{2} + 81}$
Answer
Solution:
Step:1 Identify the values that cause the function $\frac{t - 9}{t^{2} + 81}$ to be undefined. The domain is the set of all real numbers except for these values. In this instance, the denominator never equals zero, so the function is defined for all real numbers.
Step:2 Determine if there are any vertical asymptotes by looking for points of infinite discontinuity. There are no vertical asymptotes for this function.
Step:3 Analyze the rational function $R(t) = \frac{at^{n}}{bt^{m}}$ 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 $n < m$, then the horizontal asymptote is the line $y = 0$.
If $n = m$, then the horizontal asymptote is the line $y = \frac{a}{b}$.
If $n > m$, there are no horizontal asymptotes; instead, there may be an oblique asymptote.
Step:4 Calculate the degrees $n$ and $m$ for the given function.
$n = 1$ $m = 2$
Step:5 Given that $n < m$, the horizontal asymptote is the x-axis, which is the line $y = 0$.
Step:6 An oblique asymptote does not exist since the degree of the numerator is not greater than the degree of the denominator.
Step:7 Compile the list of asymptotes for the function:
No Vertical Asymptotes Horizontal Asymptote: $y = 0$ No Oblique Asymptotes
Step:8
Knowledge Notes:
The process of finding asymptotes for a rational function involves several steps and knowledge of different types of asymptotes:
Vertical Asymptotes: These occur at values of $x$ that cause the denominator of a rational function to be zero (and the numerator is not zero at those points), leading to undefined points in the function. To find vertical asymptotes, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These are horizontal lines that the graph of the function approaches as $x$ goes to infinity or negative infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator ($n$) and denominator ($m$):
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 (Slant) Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. To find an oblique asymptote, divide the numerator by the denominator using polynomial long division or synthetic division.
Rational Functions: A rational function is a function that can be expressed as the ratio of two polynomials. It is written in the form $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.
Domain of a Function: The domain of a function is the set of all possible input values (usually $x$ values) for which the function is defined. For rational functions, the domain excludes values that make the denominator zero.
In the given problem, since the degree of the numerator is less than the degree of the denominator, the function has a horizontal asymptote at $y = 0$ and no vertical or oblique asymptotes.
