Find the Asymptotes f(x)=(4x^3+15x^2+9x)/(x^3+3x^2)
In the given problem, you are tasked with determining the asymptotes of the function f(x) = (4x^3 + 15x^2 + 9x) / (x^3 + 3x^2). An asymptote refers to a line that the graph of the function approaches but never actually reaches as the input (x) either increases without bound (towards positive or negative infinity) or decreases without bound.
For rational functions like the one provided, there are typically two kinds of asymptotes to consider:
Vertical asymptotes are found where the denominator of the function equals zero, and the numerator is not zero at those points. These represent the values of x for which the function is undefined and the graph heads off to infinity.
Horizontal or oblique (slant) asymptotes relate to the behavior of the function as x approaches infinity or negative infinity. A horizontal asymptote may exist if the degrees of the numerator and denominator are equal, while an oblique asymptote may be present if the degree of the numerator is exactly one more than the denominator.
The question requires analyzing the given function's behavior as x approaches both infinity and the values that make the denominator zero to identify any vertical, horizontal, or oblique asymptotes.
$f \left(\right. x \left.\right) = \frac{4 x^{3} + 15 x^{2} + 9 x}{x^{3} + 3 x^{2}}$
Answer
Solution:
Step 1:
Determine the points where the function $\frac{4x^3 + 15x^2 + 9x}{x^3 + 3x^2}$ is not defined. These points are $x = -3$ and $x = 0$.
Step 2:
Analyze the behavior of the function as $x$ approaches $0$. As $x \to 0^-$, the function tends towards $-\infty$, and as $x \to 0^+$, it tends towards $\infty$. Thus, $x = 0$ is a vertical asymptote.
Step 3:
Examine the degrees of the polynomial in the numerator ($n$) and the denominator ($m$) in a general rational function $R(x) = \frac{ax^n}{bx^m}$ to determine the horizontal asymptote:
If $n < m$, the horizontal asymptote is $y = 0$.
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$.
If $n > m$, there is no horizontal asymptote, but there may be an oblique asymptote.
Step 4:
Identify the values of $n$ and $m$ for the given function. Here, $n = 3$ and $m = 3$.
Step 5:
Since $n$ is equal to $m$, the horizontal asymptote is found using $y = \frac{a}{b}$, where $a = 4$ and $b = 1$. Therefore, the horizontal asymptote is $y = 4$.
Step 6:
No oblique asymptote exists for this function because the degree of the numerator is not greater than the degree of the denominator.
Step 7:
Compile the list of asymptotes for the function:
- Vertical Asymptotes: $x = 0$
- Horizontal Asymptotes: $y = 4$
- No Oblique Asymptotes
Knowledge Notes:
Asymptotes are lines that a graph of a function approaches as the independent variable (usually $x$) either goes to infinity or to a point where the function is not defined. There are three types of asymptotes:
Vertical Asymptotes: These occur at values of $x$ where the function is undefined due to a zero in the denominator. The function will approach infinity or negative infinity as it gets close to these points from one side or the other.
Horizontal Asymptotes: These occur when the degrees of the numerator and denominator polynomials are the same or when the degree of the numerator is less than that of the denominator. They represent the value that the function approaches as $x$ goes to positive or negative infinity.
Oblique (Slant) 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.
In the given function $\frac{4x^3 + 15x^2 + 9x}{x^3 + 3x^2}$, we can see that the degrees of the numerator and denominator are both 3, which means there will be a horizontal asymptote and no oblique asymptote. The vertical asymptote occurs where the denominator is zero and the function is undefined.
