Find the Asymptotes f(x)=(4x)/(24x-5)

Solution:

Step 1:

Determine the value of $x$ where the function $\frac{4x}{24x-5}$ does not exist, which is when the denominator equals zero. Solve $24x-5=0$ for $x$ to find the vertical asymptote.

$x=\frac{5}{24}$

Step 2:

Examine the general form of a rational function $f(x)=\frac{a{x}^n}{b{x}^m}$, where $n$ is the degree of the polynomial in the numerator and $m$ is the degree of the polynomial in the denominator. 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; instead, there may be an oblique asymptote.

Step 3:

Identify the degrees $n$ and $m$ for the given function.

$n=1$ $m=1$

Step 4:

Since the degrees are equal ($n=m$), calculate the horizontal asymptote using the coefficients $a$ and $b$ from the leading terms of the numerator and denominator.

$y=\frac{a}{b}$ where $a=4$ and $b=24$.

$y=\frac{1}{6}$

Step 5:

Conclude that there are no oblique asymptotes for this function, as 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: $x=\frac{5}{24}$ Horizontal Asymptote: $y=\frac{1}{6}$ No Oblique Asymptotes

Step 7:

Knowledge Notes:

Asymptotes are lines that a graph of a function approaches but never touches. There are three types of asymptotes: vertical, horizontal, and oblique.

1. Vertical Asymptotes: These occur at values of $x$ where the function is undefined, typically where the denominator of a rational function is zero. They are found by setting the denominator equal to zero and solving for $x$.

2. Horizontal Asymptotes: These occur when the value of a function approaches a constant as $x$ approaches infinity or negative infinity. For a rational function $\frac{a{x}^n}{b{x}^m}$:

   • 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 in a rational function. They are found by performing polynomial long division or synthetic division.

4. Rational Functions: A function of the form $f(x)=\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, is called a rational function. The behavior of its graph near the asymptotes is an important feature of the function.

5. Degrees of Polynomials: The degree of a polynomial is the highest power of $x$ that appears in the polynomial with a non-zero coefficient. The degree influences the end behavior of the function and the possible number and type of asymptotes.

6. Leading Coefficients: The leading coefficient of a polynomial is the coefficient of the term with the highest power of $x$. It plays a crucial role in determining the equation of the horizontal asymptote for rational functions when the degrees of the numerator and denominator are equal.