Find the Asymptotes f(x)=(x^2-64)/(x+8)
The question asks for the identification of the asymptotes of the function f(x) = (x^2 - 64)/(x + 8). Here, an asymptote refers to a line that the graph of the function approaches but never actually reaches as the value of x approaches either infinity or negative infinity, or as x approaches a specific value for which the function is undefined. The question implies that there may be both vertical and horizontal asymptotes to find, and it requires analyzing the algebraic form of the function to determine at which values of x the function is undefined (for vertical asymptotes), and the end behavior of the function as x tends toward infinity or negative infinity (for horizontal or oblique asymptotes).
$f \left(\right. x \left.\right) = \frac{x^{2} - 64}{x + 8}$
Answer
Solution:
Step 1:
Determine where the function $\frac{x^{2} - 64}{x + 8}$ is not defined: $x = -8$
Step 2:
Identify any vertical asymptotes by finding values that cause the function to approach infinity. There are no vertical asymptotes in this case.
Step 3:
Examine the rational function $R(x) = \frac{a x^{n}}{b x^{m}}$ to determine horizontal asymptotes based on the degrees of the polynomial in the numerator ($n$) and the denominator ($m$):
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 there may be an oblique asymptote.
Step 4:
Calculate the degrees of the numerator and denominator: $n = 2$, $m = 1$.
Step 5:
Since $n > m$, we conclude that there are no horizontal asymptotes.
Step 6:
Determine the oblique asymptote by performing polynomial division.
Step 6.1:
Simplify the given expression.
Step 6.1.1:
First, simplify the numerator.
Step 6.1.1.1:
Express $64$ as $8^{2}$ to get $\frac{x^{2} - 8^{2}}{x + 8}$.
Step 6.1.1.2:
Use the difference of squares formula, $a^{2} - b^{2} = (a + b)(a - b)$, where $a = x$ and $b = 8$, to factor the numerator as $(x + 8)(x - 8)$.
Step 6.1.2:
Eliminate the common factor in the numerator and denominator.
Step 6.1.2.1:
Remove the common factor to get $\frac{(x + 8)(x - 8)}{x + 8}$.
Step 6.1.2.2:
After canceling the common factor, we are left with $x - 8$.
Step 6.2:
The quotient from the division, $x - 8$, represents the oblique asymptote.
Step 7:
Compile the complete set of asymptotes for the function:
- No Vertical Asymptotes
- No Horizontal Asymptotes
- Oblique Asymptote: $y = x - 8$
Knowledge Notes:
The process of finding asymptotes for a rational function involves several key concepts:
Undefined Points: A rational function is undefined where its denominator equals zero. These points can be potential vertical asymptotes.
Vertical Asymptotes: These occur at values of $x$ where the function approaches infinity. They correspond to the zeroes of the denominator that do not cancel with zeroes in the numerator.
Horizontal Asymptotes: These are horizontal lines that the graph of the function approaches as $x$ goes to infinity or negative infinity. They are determined by comparing the degrees of the numerator and denominator polynomials ($n$ and $m$):
If $n < m$, the horizontal asymptote is $y = 0$.
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.
If $n > m$, there are no horizontal asymptotes.
Oblique (Slant) Asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator ($n = m + 1$), the function may have an oblique asymptote. This is found by performing polynomial long division on the function.
Difference of Squares: The difference of squares is a pattern where $a^{2} - b^{2} = (a + b)(a - b)$. This is often used to factor expressions in the numerator or denominator of a rational function.
Polynomial Division: Polynomial long division is used to simplify rational functions and to find oblique asymptotes. The quotient (without the remainder) from the division process gives the equation of the oblique asymptote.
Cancelling Common Factors: When a common factor exists in both the numerator and the denominator, it can be cancelled out. This is important for simplifying the function and determining the correct asymptotes.
