Find the Asymptotes f(x)=(2x-5)/(x-6)
The question asks for the identification of the asymptotes of the given rational function f(x) = (2x - 5)/(x - 6). An asymptote is a line that the graph of a function approaches but never actually reaches as the input (in this case, x) either increases or decreases towards infinity.
There are typically two types of asymptotes to consider for a rational function:
Vertical asymptotes, which occur where the denominator of the function approaches zero (since division by zero is undefined). These can be found by setting the denominator equal to zero and solving for the variable.
Horizontal or oblique/slant asymptotes, which are the lines that the function approaches as x tends towards positive or negative infinity. The existence and equation of these depend on the degrees of the numerator and denominator polynomials.
The question does not require you to calculate the actual equations of the asymptotes but simply to identify them based on the given function.
$f \left(\right. x \left.\right) = \frac{2 x - 5}{x - 6}$
Answer
Solution:
Step 1:
Determine the value of $x$ that causes $\frac{2x - 5}{x - 6}$ to be undefined, which is $x = 6$.
Step 2:
Examine 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.
When $n < m$, the horizontal asymptote is the x-axis, or $y = 0$.
When $n = m$, the horizontal asymptote is the line $y = \frac{a}{b}$.
When $n > m$, there is no horizontal asymptote, but there may be an oblique asymptote.
Step 3:
Identify the values of $n$ and $m$ for the given function.
- $n = 1$
- $m = 1$
Step 4:
Since $n$ is equal to $m$, calculate the horizontal asymptote using the formula $y = \frac{a}{b}$, where $a = 2$ and $b = 1$.
- $y = 2$
Step 5:
Conclude that there is no oblique asymptote since the degree of the numerator is not greater than the degree of the denominator.
Step 6:
Summarize the asymptotes of the function.
- Vertical Asymptotes: $x = 6$
- Horizontal Asymptotes: $y = 2$
- No Oblique Asymptotes
Step 7:
Knowledge Notes:
To find the asymptotes of a rational function, one must first understand the different types of asymptotes:
Vertical Asymptotes: These occur at values of $x$ where the function is undefined, typically where the denominator is zero. To find vertical asymptotes, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These are horizontal lines that the graph of the function approaches as $x$ goes to infinity or negative infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator ($n$) and the 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.
Oblique (Slant) Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. To find an oblique asymptote, divide the numerator by the denominator using polynomial long division or synthetic division.
In the given problem, the function $f(x) = \frac{2x - 5}{x - 6}$ has a vertical asymptote at $x = 6$ because the function is undefined when $x = 6$. The degrees of the numerator and denominator are both 1 ($n = m = 1$), so the horizontal asymptote is found using the leading coefficients of the numerator and denominator, resulting in $y = \frac{2}{1} = 2$. Since the degree of the numerator is not greater than the degree of the denominator, there are no oblique asymptotes.
