Find the Asymptotes f(x)=(x^3-8x^2+15x)/(x^2-5x)
In this problem, you are asked to find the asymptotes of the given rational function f(x) = (x^3 - 8x^2 + 15x) / (x^2 - 5x). Asymptotes are lines that the graph of a function approaches as the independent variable, x in this case, goes to infinity or to a certain point where the function is undefined.
The problem involves two types of asymptotes:
Vertical asymptotes, which occur where the denominator of the rational function is equal to zero (provided the numerator isn't zero for the same values), because division by zero is undefined. This indicates where the function "blows up" and the graph goes to infinity as it approaches the line x = a.
Horizontal or oblique (slant) asymptotes, which describe the end behavior of the function as x approaches infinity or negative infinity. For rational functions:
A horizontal asymptote is determined by comparing the degrees of the polynomials in the numerator and the denominator.
If the degree of the numerator is one more than the degree of the denominator, the function will have a slant (oblique) asymptote, which is found by long division of the polynomials.
$f \left(\right. x \left.\right) = \frac{x^{3} - 8 x^{2} + 15 x}{x^{2} - 5 x}$
Answer
Solution:
Step 1: Identify Undefined Points
Determine the values for which the function $\frac{x^{3} - 8x^{2} + 15x}{x^{2} - 5x}$ is not defined. These are $x = 0$ and $x = 5$.
Step 2: Vertical Asymptotes
Check for infinite discontinuities to find vertical asymptotes. There are none in this case.
Step 3: Horizontal Asymptotes Rules
For 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:
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 potentially an oblique asymptote.
Step 4: Determine Degrees
Calculate the degrees of the numerator and denominator: $n = 3$ and $m = 2$.
Step 5: Horizontal Asymptote Conclusion
Since $n > m$, there is no horizontal asymptote.
Step 6: Oblique Asymptote Calculation
Perform polynomial long division to find the oblique asymptote.
Step 6.1: Simplify the Function
Begin simplifying the function.
Step 6.1.1: Simplify the Numerator
Start with the numerator $x^{3} - 8x^{2} + 15x$.
Step 6.1.1.1: Factor Out $x$
Extract $x$ from each term to get $x(x^{2} - 8x + 15)$.
Step 6.1.1.2: Factor the Quadratic
Use the AC method to factor $x^{2} - 8x + 15$.
Step 6.1.1.2.1: Find Integers for AC Method
Identify integers with a product of $15$ and a sum of $-8$, which are $-5$ and $-3$.
Step 6.1.1.2.2: Write Factored Form
Express the factored form as $x((x - 5)(x - 3))$.
Step 6.1.2: Simplify the Denominator
Now, focus on the denominator $x^{2} - 5x$.
Step 6.1.2.1: Factor Out $x$
Factor $x$ from the denominator to get $x(x - 5)$.
Step 6.1.2.2: Cancel Common Factors
Cancel the common $x$ factor.
Step 6.1.2.2.1: Perform Cancellation
After canceling, the function simplifies to $\frac{(x - 5)(x - 3)}{x - 5}$.
Step 6.1.2.3: Cancel Additional Common Factors
Cancel the common $(x - 5)$ factor.
Step 6.1.2.3.1: Final Simplification
The final simplified form is $x - 3$.
Step 6.2: Determine the Oblique Asymptote
The oblique asymptote is given by the simplified polynomial, which is $y = x - 3$.
Step 7: Asymptotes Summary
Summarize all asymptotes found:
- No Vertical Asymptotes
- No Horizontal Asymptotes
- Oblique Asymptote: $y = x - 3$
Step 8: End of Solution Process
Knowledge Notes:
The process involves several key concepts in algebra and calculus:
Undefined Points: Points where the function is not defined due to division by zero.
Vertical Asymptotes: Lines where the function approaches infinity. They occur at the undefined points unless there is a common factor in the numerator and denominator that cancels out.
Horizontal Asymptotes: Lines that the graph of the function approaches as $x$ goes to infinity or negative infinity. They depend on the degrees of the polynomial in the numerator ($n$) and denominator ($m$).
Oblique (Slant) Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. They can be found by performing polynomial long division.
Polynomial Long Division: A method used to divide polynomials, similar to long division with numbers.
Factoring: The process of breaking down a composite number or polynomial into a product of other numbers or polynomials that, when multiplied, give the original number or polynomial.
AC Method: A factoring technique used to factor trinomials.
Cancellation: The process of reducing a fraction by eliminating common factors from the numerator and the denominator.
