Find the Asymptotes f(x)=(4x)/(24x-5)
The problem at hand is requiring the identification of any asymptotes associated with the function f(x) = (4x)/(24x - 5). To clarify this type of question, the seeker is being asked to determine the lines that the graph of the given rational function approaches as the input (x) approaches either positive infinity, negative infinity, or some specific value at which the function is undefined. The two types of asymptotes that can occur for this function are horizontal asymptotes, describing the behavior as x grows large (both positive and negative), and vertical asymptotes, indicating values of x where the function is undefined due to a zero in the denominator. The question does not require finding any other characteristic of the function, such as its intercepts, domain, range, or critical points; it is specific to asymptote identification.
$f \left(\right. x \left.\right) = \frac{4 x}{24 x - 5}$
Answer
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 $f(x) = \frac{ax^n}{bx^m}$, 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:
If $n < m$, the horizontal asymptote is the x-axis, or $y = 0$.
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:
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.
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 $x$.
Horizontal Asymptotes: These occur when the value of a function approaches a constant as $x$ approaches infinity or negative infinity. For a rational function $\frac{ax^n}{bx^m}$:
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.
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.
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.
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.
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.
