Find the Asymptotes f(x)=(12x)/(21x-2)

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 $R(x)=\frac{a{x}^n}{b{x}^m}$, 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:

1. If $n<m$, the horizontal asymptote is the x-axis, or $y=0$.

2. If $n=m$, the horizontal asymptote is the line $y=\frac{a}{b}$.

3. 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 $R(x)=\frac{a{x}^n}{b{x}^m}$:

• 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.