Find the Asymptotes f(x)=(6x-7)/(x-7)
The given problem is asking to identify the asymptotes of the function f(x) = (6x-7)/(x-7). Asymptotes are lines that a graph approaches as x or y goes to infinity but never actually reaches. In this context, there might be a vertical asymptote associated with the value that makes the denominator zero, if that does not coincide with a zero in the numerator, and a horizontal or oblique (slant) asymptote that describes the end behavior of the function as x goes to positive or negative infinity. The question requires an examination of the function to determine if these asymptotes exist and to describe them analytically.
$f \left(\right. x \left.\right) = \frac{6 x - 7}{x - 7}$
Answer
Solution:
Step 1:
Identify the value of $x$ that causes $\frac{6x - 7}{x - 7}$ to be undefined, which is $x = 7$.
Step 2:
Examine the general form of a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ is the highest power in the numerator and $m$ is the highest power in 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 the given function. Here, $n = 1$ and $m = 1$.
Step 4:
Since $n$ equals $m$, the horizontal asymptote is found using $y = \frac{a}{b}$. For our function, $a = 6$ and $b = 1$, so the horizontal asymptote is $y = 6$.
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 Asymptotes: $x = 7$
- Horizontal Asymptotes: $y = 6$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, you need to understand the following concepts:
Undefined Points: A rational function is undefined where its denominator is zero. These points often indicate vertical asymptotes.
Horizontal Asymptotes: These are determined by comparing the degrees of the numerator ($n$) and the denominator ($m$) of the function in its highest power form. There are three cases:
If $n < m$, the horizontal asymptote is the x-axis, $y = 0$.
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: If the degree of the numerator is exactly one more than the degree of the denominator, the function may have an oblique (slant) asymptote, which can be found by polynomial long division.
Vertical Asymptotes: These occur at values of $x$ that make the denominator zero (and are not canceled by the numerator). They represent the values where the function goes to infinity.
Rational Functions: A rational function is a 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.
In the given problem, we apply these concepts to find the vertical and horizontal asymptotes of the function $f(x) = \frac{6x - 7}{x - 7}$. Since the degrees of the numerator and denominator are equal, we have a horizontal asymptote at $y = \frac{a}{b}$. There are no oblique asymptotes because the degree of the numerator is not greater than the degree of the denominator. The vertical asymptote is found by setting the denominator equal to zero and solving for $x$.
