Find the Asymptotes f(x)=(1-6x)/(1+7x)

Solution:

Step 1:

Determine the points at which the function $\frac{1-6x}{1+7x}$ does not exist. This occurs when $x=-\frac{1}{7}$.

Step 2:

Examine the general form of a rational function $R(x)=\frac{a{x}^n}{b{x}^m}$, where $n$ and $m$ are the degrees of the polynomial in the numerator and denominator, respectively.

• A horizontal asymptote at $y=0$ exists if $n<m$.

• If $n=m$, the horizontal asymptote is found at $y=\frac{a}{b}$.

• When $n>m$, 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. We find $n=1$ and $m=1$.

Step 4:

Since $n$ equals $m$, we have a horizontal asymptote at $y=\frac{a}{b}$, with $a=-6$ and $b=7$. Thus, the horizontal asymptote is $y=-\frac{6}{7}$.

Step 5:

An oblique asymptote is not present since 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: $x=-\frac{1}{7}$
• Horizontal Asymptote: $y=-\frac{6}{7}$
• No Oblique Asymptotes

Knowledge Notes:

To solve for the asymptotes of a rational function, one must understand the following key concepts:

1. Undefined Points: A rational function is undefined where its denominator equals zero. These points are potential vertical asymptotes.

2. Degrees of Polynomials: The degree of a polynomial is the highest power of $x$ in its expression. The degrees of the numerator and denominator polynomials are denoted by $n$ and $m$, respectively.

3. Horizontal Asymptotes: These occur depending on the relationship between $n$ and $m$:

   • 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.

4. Oblique Asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator ($n=m+1$), the function may have an oblique (or slant) asymptote, which can be found using polynomial long division or synthetic division.

5. Vertical Asymptotes: These are found by setting the denominator equal to zero and solving for $x$, provided that the numerator does not also equal zero at those points (which would indicate a hole in the graph rather than an asymptote).

By applying these concepts to the given function, one can determine the presence and equations of any vertical, horizontal, or oblique asymptotes.