Find the Asymptotes f(x)=(x^2-10x+21)/(x^2-5x+6)
The question asks for the identification of the asymptotes of the given function, which is a rational function represented by f(x)=(x^2-10x+21)/(x^2-5x+6). Asymptotes are lines that the graph of the function approaches but does not reach. In particular, this question seems to be asking for both vertical and horizontal asymptotes.
Vertical asymptotes occur where the denominator of the rational function equals zero, provided that the numerator does not also equal zero at those points. Finding vertical asymptotes typically involves setting the denominator equal to zero and solving for x.
Horizontal asymptotes are determined by the behavior of the function as x approaches infinity or negative infinity. They are found by comparing the degrees of the polynomials in the numerator and the denominator or by applying limits if the degrees are the same.
The question thus wants you to perform the necessary algebraic manipulations to find the x-values where the function is undefined (vertical asymptotes) and to analyze the behavior of the function at infinity (horizontal asymptotes).
$f \left(\right. x \left.\right) = \frac{x^{2} - 10 x + 21}{x^{2} - 5 x + 6}$
Answer
Solution:
Step 1:
Determine the values of $x$ for which the function $\frac{x^{2} - 10x + 21}{x^{2} - 5x + 6}$ does not exist. These are $x = 2$ and $x = 3$.
Step 2:
Observe the behavior of the function as $x$ approaches 2. As $x$ approaches 2 from the left, the function tends towards positive infinity, and from the right, it tends towards negative infinity. Therefore, $x = 2$ is a vertical asymptote.
Step 3:
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. The horizontal asymptote can be determined 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 4:
Calculate the degrees $n$ and $m$ for the numerator and denominator, respectively. In this case, $n = 2$ and $m = 2$.
Step 5:
Since the degrees of the numerator and denominator are equal ($n = m$), the horizontal asymptote is the line $y = \frac{a}{b}$. Here, $a = 1$ and $b = 1$, so $y = 1$.
Step 6:
Determine the presence of an oblique asymptote. Since the degree of the numerator is not greater than the degree of the denominator ($n \leq m$), there are no oblique asymptotes.
Step 7:
Compile the list of all asymptotes for the function:
- Vertical Asymptotes: $x = 2$
- Horizontal Asymptotes: $y = 1$
- No Oblique Asymptotes
Knowledge Notes:
The concept of asymptotes is related to the behavior of a function as it approaches a certain line or point. There are three types of asymptotes commonly discussed:
Vertical Asymptotes: These occur where the function tends towards infinity or negative infinity as the input approaches a certain value. They are found by setting the denominator of a rational function equal to zero and solving for $x$.
Horizontal Asymptotes: These describe the behavior of a function as $x$ goes to positive or negative infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator and denominator of the rational function.
Oblique (Slant) Asymptotes: These occur when the degree of the numerator is exactly one greater than the degree of the denominator. They describe a line that the graph of the function approaches but does not cross as $x$ goes to infinity or negative infinity.
In the given problem, we are dealing with a rational function where the degrees of the numerator and denominator are equal. This means that we can expect a horizontal asymptote based on the leading coefficients of the numerator and denominator. Vertical asymptotes are found by determining the values for which the function is undefined (where the denominator is zero). Oblique asymptotes are not present in this case because the degree of the numerator is not greater than that of the denominator.
