Find the Asymptotes (4x-3)/x

Solution:

Step 1:

Identify points where the function $\frac{4x-3}{x}$ does not exist. This occurs when $x = 0$.

Step 2:

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

Determine the values of $n$ and $m$. For our function, $n = 1$ and $m = 1$.

Step 4:

Since $n$ equals $m$, we find the horizontal asymptote using the formula $y = \frac{a}{b}$. In this case, $a = 4$ and $b = 1$, so the horizontal asymptote is $y = 4$.

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 Asymptotes: $x = 0$
• Horizontal Asymptotes: $y = 4$
• No Oblique Asymptotes

Knowledge Notes:

To find the asymptotes of a rational function, we need to understand the behavior of the function as $x$ approaches infinity or negative infinity, and when the function is undefined. Here are the relevant knowledge points:

1. Vertical Asymptotes: These occur at values of $x$ where the function is undefined, typically where the denominator is zero. To find vertical asymptotes, set the denominator equal to zero and solve for $x$.

2. Horizontal Asymptotes: These are found by comparing the degrees of the numerator and denominator polynomials ($n$ and $m$ respectively). The rules for finding horizontal asymptotes depend on the relationship between $n$ 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.

3. Oblique Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator ($n = m + 1$). To find an oblique asymptote, perform 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.

5. Behavior at Infinity: To determine the end behavior of a rational function, we look at what happens as $x$ approaches infinity or negative infinity. This helps us identify horizontal or oblique asymptotes.

6. Undefined Points: Points where the function is undefined (typically where the denominator is zero) are important for identifying vertical asymptotes and understanding the domain of the function.

Understanding these concepts is crucial for analyzing rational functions and finding their asymptotes.