Find the Asymptotes f(x)=(2x)/(14x-5)
The problem presented here requires identifying the asymptotes of the provided rational function f(x) = (2x)/(14x - 5).
Understanding the question involves knowing what an asymptote is: it's a line that a graph approaches but never actually reaches. Asymptotes can be vertical, horizontal, or oblique (slant).
Specifically, vertical asymptotes occur where the denominator of a rational function is zero (as long as the numerator isn't zero for the same values); horizontal asymptotes or slant asymptotes describe the behavior of the graph as x approaches infinity or negative infinity.
This question is asking for a determination of any such lines where the function f(x) grows closer and closer to but doesn't intersect with these lines for large absolute values of x or where the function is undefined due to division by zero.
$f \left(\right. x \left.\right) = \frac{2 x}{14 x - 5}$
Answer
Solution:
Step 1:
Determine the values of $x$ that cause $\frac{2x}{14x - 5}$ to be undefined, which occurs when the denominator equals zero. Solve $14x - 5 = 0$ to find $x = \frac{5}{14}$.
Step 2:
Examine the general form of a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ is the power of $x$ in the numerator and $m$ is the power of $x$ in the denominator. The horizontal asymptote rules 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 and instead, there may be an oblique asymptote.
Step 3:
Identify the values of $n$ and $m$ for the given function. Here, $n = 1$ and $m = 1$.
Step 4:
Since $n$ is equal to $m$, calculate the horizontal asymptote using the coefficients $a = 2$ and $b = 14$. The horizontal asymptote is $y = \frac{2}{14}$, which simplifies to $y = \frac{1}{7}$.
Step 5:
Confirm that there are no oblique asymptotes since 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: $x = \frac{5}{14}$
- Horizontal Asymptote: $y = \frac{1}{7}$
- No Oblique Asymptotes
Step 7:
Knowledge Notes:
To determine the asymptotes of a rational function, one must analyze the degrees of the polynomial in the numerator ($n$) and the denominator ($m$) and their respective coefficients. Here are the key points to remember:
Vertical Asymptotes: These occur at values of $x$ that make the denominator zero and thus the function undefined. To find them, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These depend on the degrees of the numerator and denominator polynomials.
If the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the horizontal asymptote is the x-axis, or $y=0$.
If $n$ equals $m$, the horizontal asymptote is found by dividing the leading coefficients of the numerator and denominator, $y = \frac{a}{b}$.
If $n$ is greater than $m$, 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 such cases, one must perform polynomial long division to find the equation of the oblique asymptote.
Nonexistence of Oblique Asymptotes: If the degree of the numerator is less than or equal to the degree of the denominator, there will be no oblique asymptotes.
In the given problem, the function $\frac{2x}{14x - 5}$ has a vertical asymptote at $x = \frac{5}{14}$ and a horizontal asymptote at $y = \frac{1}{7}$, with no oblique asymptotes due to the equal degrees of the numerator and denominator.
