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

Solution:

Step:1 Determine the values of $x$ for which the function $f(x)=\frac{3x}{6x-7}$ does not exist. This occurs when the denominator equals zero. Solve $6x-7=0$ to find $x=\frac{7}{6}$.

Step:2 Examine the general form of a rational function $R(x)=\frac{a{x}^n}{b{x}^m}$, where $n$ represents the degree of the polynomial in the numerator, and $m$ represents the degree of the polynomial in the denominator. The horizontal asymptote rules 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 $m$. For the given function, $n=1$ and $m=1$.

Step:4 Since the degrees of the numerator and denominator are equal ($n=m$), calculate the horizontal asymptote using the coefficients $a=3$ and $b=6$. The horizontal asymptote is $y=\frac{a}{b}=\frac{3}{6}=\frac{1}{2}$.

Step:5 Determine the presence of any oblique asymptotes. Since the degree of the numerator is not greater than the degree of the denominator ($n\le m$), there are no oblique asymptotes.

Step:6 Compile the complete list of asymptotes for the function:

Vertical Asymptotes: $x=\frac{7}{6}$ Horizontal Asymptotes: $y=\frac{1}{2}$ No Oblique Asymptotes

Step:7

Knowledge Notes:

1. Rational Functions: A rational function is a function that can be expressed as the quotient of two polynomials, $R(x)=\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x)\ne 0$.

2. Asymptotes: An asymptote is a line that the graph of a function approaches but never touches. There are three types of asymptotes: vertical, horizontal, and oblique (slant).

   • Vertical Asymptotes: Occur at values of $x$ where the function goes to infinity. This happens when the denominator of a rational function is zero and the numerator is non-zero.

   • Horizontal Asymptotes: Occur when the function approaches a constant value as $x$ goes to infinity or negative infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator and denominator polynomials ($n$ and $m$ respectively).

   • Oblique Asymptotes: Occur when the degree of the numerator is exactly one more than the degree of the denominator. The oblique asymptote can be found by performing polynomial long division.

3. Degrees of Polynomials: The degree of a polynomial is the highest power of $x$ that appears in the polynomial with a non-zero coefficient.

4. Undefined Points: A function is undefined at points where the denominator is zero, as division by zero is not allowed in mathematics.

5. LaTeX Formatting: LaTeX is a typesetting system commonly used for mathematical and scientific documents. It allows for the creation of complex formulas and symbols in a consistent and readable format.