Find the Asymptotes f(x)=(x+1)/(2x^2-3x-5)
The question asks for the determination of the asymptotes of the given function f(x) = (x+1)/(2x^2-3x-5). An asymptote is a line that the graph of a function approaches but never actually reaches as the independent variable (in this case, x) goes towards infinity or minus infinity. There can be both vertical and horizontal (or in some cases, oblique) asymptotes.
For vertical asymptotes, the common approach is to look for values of x that make the denominator zero, as these are the points where the function may tend to infinity. For the horizontal or oblique asymptotes, the usual method is to examine the behavior of the function as x approaches infinity or minus infinity; this often involves comparing the degrees of the polynomial in the numerator to that in the denominator.
$f \left(\right. x \left.\right) = \frac{x + 1}{2 x^{2} - 3 x - 5}$
Answer
Solution:
Step 1:
Determine the values of $x$ for which the function $\frac{x + 1}{2x^2 - 3x - 5}$ is not defined. These are $x = -1$ and $x = \frac{5}{2}$.
Step 2:
Examine the behavior of $\frac{x + 1}{2x^2 - 3x - 5}$ as $x$ approaches $\frac{5}{2}$ from both sides. Since the function approaches $-\infty$ from the left and $\infty$ from the right, we establish that $x = \frac{5}{2}$ is a vertical asymptote.
Step 3:
Review the general form of a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ is the highest power of $x$ in the numerator and $m$ is the highest power of $x$ in the denominator. The rules for horizontal asymptotes are as follows:
If $n < m$, the horizontal asymptote is the x-axis, $y = 0$.
If $n = m$, the horizontal asymptote is the line $y = \frac{a}{b}$.
If $n > m$, there are no horizontal asymptotes; instead, there may be an oblique asymptote.
Step 4:
Identify the degrees $n$ and $m$ for the given function. Here, $n = 1$ and $m = 2$.
Step 5:
Since the degree of the numerator $n$ is less than the degree of the denominator $m$, the horizontal asymptote is the x-axis, which is the line $y = 0$.
Step 6:
Conclude that there are no oblique asymptotes because the degree of the numerator is not greater than the degree of the denominator.
Step 7:
Compile the list of all asymptotes for the function:
- Vertical Asymptotes: $x = \frac{5}{2}$
- Horizontal Asymptotes: $y = 0$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, one must understand the following concepts:
Vertical Asymptotes: These occur at values of $x$ where the denominator of the rational function is zero, and the numerator is not zero at those points. The function will approach infinity or negative infinity near these points.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and denominator. If the degree of the numerator ($n$) is less than the degree of the denominator ($m$), 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. The oblique asymptote can be found by performing polynomial long division.
Undefined Points: The function is undefined where the denominator equals zero. These points can lead to vertical asymptotes unless the numerator is also zero at these points, which would indicate a hole in the graph.
Behavior at Infinity: To determine the behavior of the function as $x$ approaches infinity or negative infinity, one can look at the leading terms of the numerator and denominator.
Understanding these concepts is crucial for analyzing the behavior of rational functions and identifying their asymptotes.
