Find the Asymptotes (3x)/(x+4)
The given problem is asking to determine the asymptotes of the rational function (3x)/(x+4). Asymptotes are lines that the graph of a function approaches but does not actually reach as the values go to infinity or negative infinity. The question requires you to find both the vertical and horizontal asymptotes, if any, for this particular rational function. Vertical asymptotes are found by determining where the function is undefined (i.e., where the denominator equals zero), while horizontal asymptotes are found by considering the behavior of the function as x approaches positive or negative infinity.
$\frac{3 x}{x + 4}$
Answer
Solution:
Step 1:
Identify the values for which the function $\frac{3x}{x+4}$ is not defined. This occurs when the denominator equals zero, which is at $x = -4$.
Step 2:
Examine the degrees of the numerator and denominator in a rational function $R(x) = \frac{ax^n}{bx^m}$. The rules for determining asymptotes are:
Horizontal asymptote at $y = 0$ if $n < m$.
Horizontal asymptote at $y = \frac{a}{b}$ if $n = m$.
No horizontal asymptote if $n > m$ (look for an oblique asymptote instead).
Step 3:
Determine the degrees of the numerator and denominator. In this case, $n = 1$ and $m = 1$.
Step 4:
Since the degrees are equal ($n = m$), the horizontal asymptote is found using $y = \frac{a}{b}$. With $a = 3$ and $b = 1$, we get $y = 3$.
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 Asymptote: $x = -4$
- Horizontal Asymptote: $y = 3$
- No Oblique Asymptote
Step 7:
Knowledge Notes:
To find the asymptotes of a rational function, you need to understand the following concepts:
Undefined Points: A rational function is undefined where its denominator is zero. These points often correspond to vertical asymptotes.
Degrees of Polynomials: The degree of a polynomial is the highest power of $x$ with a non-zero coefficient.
Horizontal Asymptotes: These are determined by comparing the degrees of the numerator ($n$) and denominator ($m$) of the rational function:
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, and one may need to look for an oblique asymptote.
Oblique Asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator, the rational function may have an oblique asymptote, which can be found using polynomial long division or synthetic division.
Vertical Asymptotes: These occur at the values of $x$ that make the denominator zero (and are not canceled out by the numerator).
Rational Functions: A rational function is a function that can be expressed as the quotient of two polynomials.
By applying these concepts, one can systematically find the vertical and horizontal asymptotes of a given rational function. Oblique asymptotes require additional steps if the degree of the numerator exceeds that of the denominator by one.
