Find the Asymptotes f(x)=x/(4-x^2)
The question is asking for the identification of the asymptotes of the given function f(x) = x / (4 - x^2). Asymptotes are lines that the graph of the function approaches but never actually touches or reaches as x approaches certain values. There are typically two types of asymptotes to consider for rational functions: vertical and horizontal (or sometimes oblique/slant asymptotes).
Vertical asymptotes occur where the function goes to infinity, which can happen when the denominator of a fraction goes to zero (provided that the numerator doesn't also go to zero at the same point).
Horizontal or slant asymptotes describe the behavior of the graph as x goes to positive or negative infinity, giving an idea of how the function behaves in the far left and right ends of the graph.
This involves analyzing the function to determine where these conditions occur and describing the asymptotes with equations in the form of y = mx + b for slant asymptotes, y = a (a constant) for horizontal asymptotes, or x = c (a constant) for vertical asymptotes.
$f \left(\right. x \left.\right) = \frac{x}{4 - x^{2}}$
Answer
Solution:
Step 1:
Determine the values for which the function $\frac{x}{4 - x^2}$ does not exist. These are $x = -2$ and $x = 2$.
Step 2:
Observe the behavior of $\frac{x}{4 - x^2}$ as $x$ approaches $-2$ from the left and right. The function tends towards positive and negative infinity, respectively. Hence, $x = -2$ is a vertical asymptote.
Step 3:
Similarly, as $x$ approaches $2$ from the left and right, $\frac{x}{4 - x^2}$ heads towards positive and negative infinity, respectively. This indicates that $x = 2$ is also a vertical asymptote.
Step 4:
Compile a list of the vertical asymptotes: $x = -2$ and $x = 2$.
Step 5:
For a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ and $m$ are the degrees of the numerator and denominator respectively:
If $n < m$, the horizontal asymptote is the x-axis, $y = 0$.
If $n = m$, the horizontal asymptote is the line $y = \frac{a}{b}$.
If $n > m$, there is no horizontal asymptote, but possibly an oblique asymptote.
Step 6:
Identify the degrees $n$ and $m$ for the given function. Here, $n = 1$ and $m = 2$.
Step 7:
Since the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the horizontal asymptote is the x-axis, given by $y = 0$.
Step 8:
An oblique asymptote is ruled out because the degree of the numerator is not greater than the degree of the denominator.
Step 9:
Summarize the asymptotes of the function:
Vertical Asymptotes: $x = -2$, $x = 2$ Horizontal Asymptotes: $y = 0$ No Oblique Asymptotes
Knowledge Notes:
Vertical Asymptotes: These occur at the values of $x$ where the function becomes undefined, typically where the denominator is zero. The function's value will approach infinity or negative infinity near these points.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and the denominator ($n$ and $m$) in a rational function. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$. If the degrees are equal, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.
Oblique 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.
Behavior Near Asymptotes: As $x$ approaches the vertical asymptote, the function's value will tend towards infinity or negative infinity. The direction (positive or negative) depends on the function's behavior on each side of the asymptote.
Rational Functions: A function of the form $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, is known as a rational function. The properties of its asymptotes are determined by the degrees and coefficients of these polynomials.
