Find the Asymptotes (X^2+2X-3)/(X-1)
The problem is asking for the identification of asymptotes for the given rational function, which is a ratio of two polynomials. An asymptote is a line that the graph of a function approaches but never actually reaches. The function provided is (X^2 + 2X - 3) / (X - 1). There are typically two types of asymptotes to be considered for such a function:
Vertical asymptotes, which occur where the denominator of the rational function is zero (since division by zero is undefined). For the given function, you would look at the values of X that make (X - 1) equal to zero.
Horizontal or oblique (slant) asymptotes, which describe the behavior of the graph as X approaches positive or negative infinity. To find these, you would commonly look at the degrees of the polynomials in the numerator and the denominator and apply certain criteria to determine the asymptotes' equations.
$\frac{X^{2} + 2 X - 3}{X - 1}$
Answer
Solution:
Step 1:
Identify the values for which the function $\frac{x^{2} + 2x - 3}{x - 1}$ is not defined. This occurs when $x = 1$.
Step 2:
Determine the locations of vertical asymptotes, which are typically where the function is undefined due to division by zero. In this case, there are no vertical asymptotes.
Step 3:
Examine the degrees of the numerator and denominator in the rational function $R(x) = \frac{ax^{n}}{bx^{m}}$. The rules for horizontal asymptotes 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 4:
Calculate the degrees of the numerator and denominator. Here, $n = 2$ and $m = 1$.
Step 5:
Since the degree of the numerator is greater than the degree of the denominator ($n > m$), there are no horizontal asymptotes.
Step 6:
To find the oblique asymptote, perform polynomial long division.
Step 6.1:
First, simplify the given expression.
Step 6.1.1:
Factor the numerator $x^{2} + 2x - 3$ using the AC method.
Step 6.1.1.1:
Look for two integers whose product equals $c$ and whose sum equals $b$. In this case, find integers with a product of $-3$ and a sum of $2$. These integers are $-1$ and $3$.
Step 6.1.1.2:
Express the factored form with these integers as $\frac{(x - 1)(x + 3)}{x - 1}$.
Step 6.1.2:
Eliminate the common factor from the numerator and denominator.
Step 6.1.2.1:
Remove the common factor to get $\frac{\cancel{(x - 1)}(x + 3)}{\cancel{x - 1}}$.
Step 6.1.2.2:
Divide the remaining polynomial $x + 3$ by $1$ to get $x + 3$.
Step 6.2:
The quotient from the division gives the equation of the oblique asymptote, which is $y = x + 3$.
Step 7:
Combine all the findings to list the asymptotes of the function. There are no vertical or horizontal asymptotes, but there is an oblique asymptote at $y = x + 3$.
Knowledge Notes:
Asymptotes are lines that a graph approaches but does not actually reach. They can be vertical, horizontal, or oblique (slant).
Vertical asymptotes occur where the function is undefined due to division by zero.
Horizontal asymptotes are determined by the degrees of the numerator ($n$) and denominator ($m$) in a rational function. If $n < m$, the horizontal asymptote is $y = 0$; if $n = m$, it is $y = \frac{a}{b}$; if $n > m$, there are no horizontal asymptotes.
Oblique asymptotes occur when the degree of the numerator is one more than the degree of the denominator. They can be found by performing polynomial long division.
Polynomial long division is a method used to divide polynomials, similar to the long division of numbers. The result can provide the equation of an oblique asymptote.
Factoring using the AC method involves finding two numbers that multiply to give the product of the coefficient of $x^2$ (A) and the constant term (C), and add up to the coefficient of $x$ (B). This is used to factor quadratic expressions.
Cancellation is the process of simplifying a fraction by eliminating common factors from the numerator and the denominator.
