Find the Asymptotes f(x)=(16x)/(20x-3)

Solution:

Step:1 Determine the value of $x$ that causes the denominator of $\frac{16x}{20x-3}$ to be zero. This is found by setting $20x - 3 = 0$ and solving for $x$. The result is $x = \frac{3}{20}$.

Step:2 Examine the degrees of the numerator and denominator in the rational function $f(x) = \frac{ax^n}{bx^m}$. The behavior of the asymptotes is determined as follows:

1. If the degree of the numerator $n$ is less than the degree of the denominator $m$, the horizontal asymptote is the x-axis, or $y = 0$.

2. If the degrees are equal, $n = m$, the horizontal asymptote is given by the ratio of the leading coefficients, $y = \frac{a}{b}$.

3. If the degree of the numerator $n$ is greater than the degree of the denominator $m$, there is no horizontal asymptote; instead, there may be an oblique asymptote.

Step:3 Calculate the degrees of the numerator and denominator. For the given function, $n = 1$ and $m = 1$.

Step:4 Since the degrees of the numerator and denominator are equal ($n = m$), the horizontal asymptote is determined by the ratio of the leading coefficients $a$ and $b$. For the given function, $a = 16$ and $b = 20$, resulting in a horizontal asymptote at $y = \frac{16}{20} = \frac{4}{5}$.

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 Summarize the asymptotes for the function:

Vertical Asymptotes: $x = \frac{3}{20}$ Horizontal Asymptotes: $y = \frac{4}{5}$ No Oblique Asymptotes

Step:7

Knowledge Notes:

The process of finding asymptotes for a rational function involves understanding the relationship between the degrees of the numerator and denominator and their leading coefficients. Here are the relevant knowledge points:

1. Rational Functions: A rational function is a function that can be expressed as the ratio of two polynomials.

2. Undefined Points: Points where the function is undefined typically occur where the denominator is zero. These points can indicate vertical asymptotes.

3. Degrees of Polynomials: The degree of a polynomial is the highest power of the variable in the polynomial.

4. Horizontal Asymptotes: These are horizontal lines that the graph of the function approaches as $x$ goes to infinity or negative infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator and denominator:

   • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.

   • If the degrees are equal, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.

   • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

5. Oblique Asymptotes: These 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.

6. Vertical Asymptotes: These are vertical lines that the graph of the function approaches as it heads towards infinity or negative infinity in the y-direction. They occur at the values of $x$ that make the denominator zero (as long as these points do not also make the numerator zero, which would instead be a hole in the graph).

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