Find the Asymptotes 8-4/x
The question is requesting to determine the asymptotes of the function f(x) = 8 - 4/x. Asymptotes are lines that the graph of a function approaches as the variable, in this case x, either goes to infinity or negative infinity or to certain critical points where the function is undefined. The problem involves finding both the vertical and horizontal asymptotes, if any, for the given rational function. Vertical asymptotes occur where the function is undefined due to a zero in the denominator, and horizontal asymptotes represent the limiting behavior of the function as x approaches positive or negative infinity.
$8 - \frac{4}{x}$
Answer
Solution:
Step 1:
Identify the values of $x$ for which $8 - \frac{4}{x}$ does not exist. This occurs when $x = 0$.
Step 2:
Examine the general form of a rational function $R(x) = \frac{a x^n}{b x^m}$, where $n$ is the highest power in the numerator and $m$ is the highest power in the denominator.
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, but there may be an oblique asymptote.
Step 3:
Determine the values of $n$ and $m$ for our function. We find $n = 0$ and $m = 1$.
Step 4:
Since $n$ equals $m$, we use the ratio $\frac{a}{b}$ to find the horizontal asymptote. For our function, $a = 8$ and $b = 1$, thus $y = \frac{8}{1} = 8$.
Step 5:
An oblique asymptote is not present because the numerator's degree is not greater than the denominator's degree.
Step 6:
Compile the list of asymptotes for the function:
- Vertical Asymptotes: $x = 0$
- Horizontal Asymptotes: $y = 8$
- No Oblique Asymptotes
Knowledge Notes:
Asymptotes are lines that a graph approaches but never actually touches or crosses. There are three types of asymptotes: vertical, horizontal, and oblique.
Vertical Asymptotes occur at values of $x$ that make the function undefined. These are typically found by setting the denominator of a rational function equal to zero and solving for $x$.
Horizontal Asymptotes are found by comparing the degrees of the polynomial in the numerator ($n$) and the denominator ($m$) of a rational function. If the degree of the numerator is less than the degree of the denominator ($n < m$), the x-axis ($y = 0$) is the horizontal asymptote. If the degrees are equal ($n = m$), the horizontal asymptote is found by dividing the leading coefficients ($y = \frac{a}{b}$). If the numerator's degree is greater ($n > m$), there are no horizontal asymptotes.
Oblique Asymptotes (also known as slant asymptotes) occur when the degree of the numerator is exactly one more than the degree of the denominator ($n = m + 1$). They can be found by performing polynomial long division or synthetic division.
In the context of the problem, the function $8 - \frac{4}{x}$ has a vertical asymptote at $x = 0$ because the function is undefined at that value. The horizontal asymptote is at $y = 8$ because the degrees of the numerator and denominator are equal, and the leading coefficient of the numerator is 8. There are no oblique asymptotes because the degree of the numerator is not greater than the degree of the denominator.
