Find the Asymptotes f(x)=(3x-7)/(x-6)

Solution:

Step 1:

Identify the values of $x$ that make the function $\frac{3x-7}{x-6}$ undefined. This occurs when the denominator equals zero: $x-6=0$, hence $x=6$.

Step 2:

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

• If $n<m$, the horizontal asymptote is the x-axis, given by $y=0$.

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

• If $n>m$, there is no horizontal asymptote; instead, there may be an oblique asymptote.

Step 3:

Determine the values of $n$ and $m$ for the given function. In this case, $n=1$ and $m=1$.

Step 4:

Since $n$ is equal to $m$, the horizontal asymptote can be found using the formula $y=\frac{a}{b}$, where $a=3$ and $b=1$. Therefore, the horizontal asymptote is $y=3$.

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 Asymptote: $x=6$
• Horizontal Asymptote: $y=3$
• There are no Oblique Asymptotes.

Step 7:

The process is complete, and the asymptotes have been identified.

Knowledge Notes:

To find the asymptotes of a rational function, one must understand the behavior of the function as $x$ approaches certain critical values. Asymptotes are lines that the graph of a function approaches but never touches.

1. Vertical Asymptotes: These occur at values of $x$ that make the denominator of a rational function zero (provided that the numerator is not also zero at those points). To find them, set the denominator equal to zero and solve for $x$.

2. Horizontal Asymptotes: These are found by comparing the degrees of the numerator and the denominator ($n$ and $m$, respectively).

   • 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.

3. 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. This is found by performing polynomial long division or synthetic division.

4. 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.

Understanding these concepts is crucial for analyzing the behavior of rational functions and identifying their asymptotes.