Find the Asymptotes f(x)=(x^2-x-20)/(x^2-10x+24)
The problem is asking for the asymptotes of the given function f(x) = (x^2 - x - 20)/(x^2 - 10x + 24). Asymptotes are lines that the graph of a function approaches as the inputs or outputs (x or f(x)) increase or decrease without bound.
There are two types of asymptotes that might be relevant for this function:
Vertical Asymptotes occur where the denominator of the fraction equals zero (as long as the numerator doesn't also equal zero at the same point, which would indicate a hole in the graph instead). They represent the values of x for which the function tends to positive or negative infinity.
Horizontal Asymptotes are investigated by examining the behavior of the function as x goes to positive or negative infinity. This typically involves looking at the leading coefficients and the degrees of the polynomials in the numerator and the denominator.
The question is directing you to determine if any such asymptotes exist for the given rational function and to identify them.
$f \left(\right. x \left.\right) = \frac{x^{2} - x - 20}{x^{2} - 10 x + 24}$
Answer
Solution:
Step 1:
Determine the values of $x$ for which the function $\frac{x^{2} - x - 20}{x^{2} - 10x + 24}$ does not exist. These are $x = 4$ and $x = 6$.
Step 2:
Observe the behavior of the function as $x$ approaches 4. As $x$ approaches 4 from the left, the function tends towards $-\infty$, and from the right, it tends towards $\infty$. Therefore, $x = 4$ is a vertical asymptote.
Step 3:
Observe the behavior of the function as $x$ approaches 6. As $x$ approaches 6 from the left, the function tends towards $-\infty$, and from the right, it tends towards $\infty$. Hence, $x = 6$ is a vertical asymptote.
Step 4:
Compile a list of all vertical asymptotes: $x = 4$ and $x = 6$.
Step 5:
Review the rules for finding horizontal asymptotes for the 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 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 are no horizontal asymptotes (but possibly an oblique asymptote).
Step 6:
Identify the degrees $n$ and $m$ of the numerator and denominator, respectively. Both are 2, so $n = 2$ and $m = 2$.
Step 7:
Since $n$ equals $m$, the horizontal asymptote is determined by the ratio of the leading coefficients $a$ and $b$. Both coefficients are 1, so the horizontal asymptote is $y = \frac{1}{1}$ or $y = 1$.
Step 8:
Conclude that there are no oblique asymptotes since the degree of the numerator is not greater than the degree of the denominator.
Step 9:
Summarize all asymptotes of the function:
- Vertical Asymptotes: $x = 4$, $x = 6$
- Horizontal Asymptote: $y = 1$
- No Oblique Asymptotes
Knowledge Notes:
Vertical Asymptotes: These occur at values of $x$ where the function becomes undefined, typically where the denominator of a rational function is zero.
Horizontal Asymptotes: These are horizontal lines that the graph of the function approaches as $x$ goes to infinity or negative infinity. The rules for determining horizontal asymptotes depend on the degrees of the numerator and denominator of the rational function.
Oblique Asymptotes: Also known as slant 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.
Rational Functions: A function defined by the ratio of two polynomials. The general form is $R(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials.
Behavior Around Asymptotes: As the function approaches a vertical asymptote, the function values typically increase or decrease without bound. As it approaches a horizontal asymptote, the function values get closer and closer to the asymptote value.
Leading Coefficients: In a polynomial, the leading coefficient is the coefficient of the term with the highest power of $x$. It plays a significant role in determining the end behavior of the function and the equation of horizontal asymptotes for rational functions.
