Find the Asymptotes f(x)=(12x)/(21x-2)
The question asks for the determination of the asymptotes of the function f(x) = (12x) / (21x - 2). Specifically, it is looking for both vertical and horizontal asymptotes, which are lines that the graph of the function approaches but never touches as the value of x approaches certain critical points or infinity. Vertical asymptotes occur at values of x that make the denominator zero, while horizontal asymptotes are determined by the end behavior of the function as x approaches infinity or negative infinity. The answer would involve analyzing the function to identify these asymptotes.
$f \left(\right. x \left.\right) = \frac{12 x}{21 x - 2}$
Answer
Solution:
Step:1 Determine the values of $x$ for which the function $f(x) = \frac{12x}{21x - 2}$ is not defined by setting the denominator equal to zero and solving for $x$. The function is undefined when $x = \frac{2}{21}$.
Step:2 Examine the general form of a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ is the power of $x$ in the numerator and $m$ is the power of $x$ in the denominator. The rules for horizontal asymptotes are as follows:
If $n < m$, the horizontal asymptote is the x-axis, or $y = 0$.
If $n = m$, the horizontal asymptote is the line $y = \frac{a}{b}$.
If $n > m$, there is no horizontal asymptote, but there may be an oblique asymptote.
Step:3 Identify the values of $n$ and $m$ in the given function. For $f(x) = \frac{12x}{21x - 2}$, we have $n = 1$ and $m = 1$.
Step:4 Since $n$ is equal to $m$, we find the horizontal asymptote using the formula $y = \frac{a}{b}$, where $a = 12$ and $b = 21$. Simplifying the fraction, we get the horizontal asymptote $y = \frac{4}{7}$.
Step:5 An oblique asymptote does not exist for this function 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: $x = \frac{2}{21}$ Horizontal Asymptotes: $y = \frac{4}{7}$ There are no Oblique Asymptotes.
Step:7
Knowledge Notes:
To determine the asymptotes of a rational function, one must understand the relationship between the degrees of the numerator and denominator. The degree of a term in a polynomial is the exponent of the variable. For a rational function $R(x) = \frac{ax^n}{bx^m}$:
Vertical asymptotes occur at values of $x$ that make the denominator zero, as the function approaches infinity or negative infinity.
Horizontal asymptotes are determined by comparing the degrees of the numerator and the denominator ($n$ and $m$):
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, but there may be an oblique asymptote, which is found using polynomial long division or other methods.
Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. They are the slant lines that the graph approaches as $x$ goes to infinity or negative infinity.
In the given problem, the function $f(x) = \frac{12x}{21x - 2}$ has a vertical asymptote at $x = \frac{2}{21}$ and a horizontal asymptote at $y = \frac{4}{7}$. There is no oblique asymptote because the degrees of the numerator and denominator are equal.
