Find the Asymptotes f(x)(x^2-8)/(2x^2-18)
The question is asking for the identification of the asymptotes of the function f(x) defined by the rational expression (x^2 - 8) / (2x^2 - 18). Asymptotes are lines that a graph approaches as the x or y values increase or decrease without bounds. There are generally two types of asymptotes that are relevant for rational functions like this one: vertical and horizontal (or oblique) asymptotes.
Vertical asymptotes occur where the denominator of the rational function is equal to zero, as long as the numerator does not also equal zero at the same points. This typically involves finding the roots of the denominator and verifying whether those points are not canceled by the numerator.
Horizontal or oblique asymptotes involve the behavior of the function as x approaches infinity or negative infinity. These usually depend on the degrees of the polynomials in the numerator and the denominator. A horizontal asymptote is determined by the ratio of the leading coefficients if the degrees of the numerator and denominator are the same, while there is no horizontal asymptote if the degree of the numerator is greater than the denominator (in which case there might be an oblique asymptote instead).
The question does not explicitly ask to differentiate between types of asymptotes, so the task is to calculate both vertical and potentially horizontal or oblique asymptotes.
$f \left(\right. x \left.\right) \frac{x^{2} - 8}{2 x^{2} - 18}$
Answer
Solution:
Identifying Asymptotes
Step 1:
The function $y = f(x) = \frac{x^2 - 8}{2x^2 - 18}$ is a rational function. To find the horizontal asymptote, we compare the degrees of the numerator and denominator. Since the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. Thus, the horizontal asymptote is $y = \frac{1}{2}$.
Step 2:
To find the vertical asymptotes, we set the denominator equal to zero and solve for $x$. The denominator $2x^2 - 18 = 0$ simplifies to $x^2 = 9$, which gives us $x = \pm 3$. Therefore, the vertical asymptotes are $x = 3$ and $x = -3$.
Knowledge Notes:
Rational functions are quotients of two polynomials. The asymptotes of a rational function can be vertical, horizontal, or oblique.
Vertical Asymptotes: These occur at values of $x$ where the denominator is zero and the numerator is not zero. To find them, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These occur when the degrees of the numerator and denominator are the same or when the degree of the numerator is less than the degree of the denominator. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is less, the horizontal asymptote is $y = 0$.
Oblique Asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator, the function may have an oblique asymptote, which is found using polynomial long division.
In the given function $y = f(x) = \frac{x^2 - 8}{2x^2 - 18}$, the degrees of the numerator and denominator are equal, leading to a horizontal asymptote. The vertical asymptotes are found by setting the denominator to zero and solving for $x$. There are no oblique asymptotes in this case because the degrees of the numerator and denominator do not differ by exactly one.
