Find the Asymptotes f(x)=(2x-1)/(15x^2-12)
The question asks for the determination of the asymptotes of the function f(x)=(2x-1)/(15x^2-12). Asymptotes are lines that the graph of a function approaches as the independent variable (in this case, x) heads towards infinity or negative infinity. There are two types of asymptotes that might occur for this function: vertical and horizontal (or potentially oblique).
Vertical asymptotes occur where the function is undefined, which typically happens at values of x where the denominator is zero (since division by zero is undefined).
Horizontal asymptotes are lines that the graph of the function approaches as x goes to infinity or negative infinity. These are found by examining the end behavior of the function.
The question requires an analysis of the function to determine where these asymptotes are located with respect to the values of x, based on algebraic properties and limits of the function.
$f \left(\right. x \left.\right) = \frac{2 x - 1}{15 x^{2} - 12}$
Answer
Solution:
Step 1:
Identify the values of $x$ for which $\frac{2x - 1}{15x^2 - 12}$ does not exist. These values are $x = -\frac{2\sqrt{5}}{5}$ and $x = \frac{2\sqrt{5}}{5}$.
Step 2:
Observe the behavior of $\frac{2x - 1}{15x^2 - 12}$ as $x$ approaches $-\frac{2\sqrt{5}}{5}$ from both sides. The function tends towards negative and positive infinity, indicating a vertical asymptote at $x = -\frac{2\sqrt{5}}{5}$.
Step 3:
Similarly, as $x$ approaches $\frac{2\sqrt{5}}{5}$ from both sides, the function again tends towards negative and positive infinity, confirming another vertical asymptote at $x = \frac{2\sqrt{5}}{5}$.
Step 4:
Compile a list of vertical asymptotes: $x = -\frac{2\sqrt{5}}{5}$ and $x = \frac{2\sqrt{5}}{5}$.
Step 5:
Consider the general form of a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ is the degree of the numerator and $m$ is the degree of the denominator. The rules for horizontal asymptotes are as follows:
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 6:
Determine the degrees $n$ and $m$ for the given function: $n = 1$ and $m = 2$.
Step 7:
Since $n < m$, the horizontal asymptote is the x-axis, which is $y = 0$.
Step 8:
There is no oblique asymptote, as the degree of the numerator ($n$) is less than the degree of the denominator ($m$).
Step 9:
Summarize all asymptotes for the function:
- Vertical Asymptotes: $x = -\frac{2\sqrt{5}}{5}$ and $x = \frac{2\sqrt{5}}{5}$
- Horizontal Asymptote: $y = 0$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, one must consider both vertical and horizontal (or oblique) asymptotes.
Vertical Asymptotes: These occur at values of $x$ where the denominator of the rational function equals zero, provided the numerator does not also equal zero at those points. The function will approach infinity or negative infinity as it gets close to these points from either side.
Horizontal Asymptotes: These are determined by comparing the degrees of the numerator ($n$) and denominator ($m$) of the rational function in its simplified form.
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. Instead, one may need to look for an oblique asymptote.
Oblique Asymptotes: These occur when the degree of the numerator is exactly one greater than the degree of the denominator. To find an oblique asymptote, one can perform polynomial long division or synthetic division to find the slant asymptote equation.
Undefined Points: When finding vertical asymptotes, it is also important to check for points where the function is undefined but does not necessarily have an asymptote, such as holes in the graph.
Behavior at Asymptotes: To confirm a vertical asymptote, one should check the limits of the function as $x$ approaches the asymptote value from both the left and right. If the function approaches infinity or negative infinity from both sides, it confirms the presence of a vertical asymptote.
In the given problem, the function $\frac{2x - 1}{15x^2 - 12}$ has two vertical asymptotes and one horizontal asymptote. There are no oblique asymptotes because the degree of the numerator is less than the degree of the denominator.
