Find the Asymptotes (x+1)/(x^2-3x-4)
The question asks to find the asymptotes of the rational function (x+1)/(x^2-3x-4). Asymptotes are lines that a graph approaches but doesn't actually touch or cross. There are typically two kinds of asymptotes to consider for such a function: vertical and horizontal (or oblique) asymptotes.
Vertical asymptotes occur where the denominator of the function goes to zero (as long as the numerator does not go to zero at the same points), because this can cause the function's value to approach infinity.
Horizontal or oblique asymptotes pertain to the end behavior of the function as x goes to positive or negative infinity, and are determined by comparing the degrees of the numerator and denominator polynomials.
The question involves performing the necessary algebraic analysis to determine these lines for the given expression.
$\frac{x + 1}{x^{2} - 3 x - 4}$
Answer
Solution:
Step 1:
Identify the values of $x$ for which $\frac{x + 1}{x^{2} - 3x - 4}$ is not defined. These are $x = -1$ and $x = 4$.
Step 2:
Observe the behavior of $\frac{x + 1}{x^{2} - 3x - 4}$ as $x$ approaches $4$. It tends towards $-\infty$ from the left and $\infty$ from the right, indicating a vertical asymptote at $x = 4$.
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 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 is no horizontal asymptote, but possibly an oblique asymptote.
Step 4:
Determine the degrees $n$ and $m$ for our function. We find $n = 1$ and $m = 2$.
Step 5:
With $n < m$, the horizontal asymptote of our function is the x-axis, given by $y = 0$.
Step 6:
An oblique asymptote is not present as the degree of the numerator ($n$) is not greater than the degree of the denominator ($m$).
Step 7:
Compile the list of asymptotes for the function:
- Vertical Asymptotes: $x = 4$
- Horizontal Asymptotes: $y = 0$
- No Oblique Asymptotes
Knowledge Notes:
Asymptotes: An asymptote is a line that a graph of a function approaches but never touches. Asymptotes can be vertical, horizontal, or oblique (slant).
Vertical Asymptotes: These occur at values of $x$ where the function is undefined and the function values approach infinity or negative infinity as $x$ approaches these points from either side.
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 and denominator of the rational function.
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.
Rational Functions: A rational function is a function that can be expressed as the quotient of two polynomials. The general form is $R(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials.
Behavior Near Asymptotes: To determine the behavior of the function near its vertical asymptotes, one should analyze the limits of the function as it approaches the asymptote values from the left and right.
Degrees of Polynomials: The degree of a polynomial is the highest power of $x$ with a non-zero coefficient. In a rational function, comparing the degrees of the numerator and the denominator helps in determining the presence and type of asymptotes.
