Find the Asymptotes (8x)/(x+2)
The problem is asking to determine the asymptotes of the rational function f(x) = (8x)/(x+2). Asymptotes are lines that the graph of a function approaches as the inputs or outputs increase or decrease without bound. Specifically, in this context, the question is likely asking for any vertical asymptotes, which occur when the function's denominator approaches zero, and horizontal asymptotes, which describe the behavior of the graph as x approaches positive or negative infinity. Identifying these lines involves analyzing the algebraic form of the function and applying the properties of limits and rational functions.
$\frac{8 x}{x + 2}$
Answer
Solution:
Step 1:
Determine the values for which the function $\frac{8x}{x+2}$ is not defined. This occurs when the denominator equals zero: $x = -2$.
Step 2:
Examine the general form of a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ is the degree of the polynomial in the numerator, and $m$ is the degree of the polynomial in the denominator. The rules for finding 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:
Identify the values of $n$ and $m$ for the given function. For $\frac{8x}{x+2}$, we have $n = 1$ and $m = 1$.
Step 4:
Since $n$ is equal to $m$, we find the horizontal asymptote using the formula $y = \frac{a}{b}$. With $a = 8$ and $b = 1$, the horizontal asymptote is $y = 8$.
Step 5:
An oblique asymptote is not present because 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: $x = -2$
- Horizontal Asymptote: $y = 8$
- No Oblique Asymptote
Step 7:
No further steps are required as all asymptotes have been identified.
Knowledge Notes:
To find the asymptotes of a rational function, one must understand the different types of asymptotes and the conditions under which they occur:
Vertical Asymptotes: These occur at values of $x$ that make the denominator of a rational function zero and the numerator non-zero. They can be found by setting the denominator equal to zero and solving 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 the 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 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.
Non-Existence of Asymptotes: If the degree of the numerator is less than or equal to the degree of the denominator and there is no oblique asymptote, then only vertical and horizontal asymptotes are considered.
In the given problem, the function $\frac{8x}{x+2}$ has a vertical asymptote at $x = -2$ and a horizontal asymptote at $y = 8$, with no oblique asymptote since the degrees of the numerator and denominator are equal.
