A rational function has the form f(x)=ax^2 / x^n -5 where a is a non-zero real number and n is a positive
integer.
a) Are there any values of a and n for which the function above will have no vertical asymptotes. Explain
why by providing examples and showing work.
b) Are there any values of a and n for which the function above will have a horizontal asymptote. Explain
why by providing examples and showing work.
(a) no, if a≠0. x^n-5 = 0 somewhere for any value of n
(b) yes -- if n=2 and a≠0
a) Vertical asymptotes occur when the denominator of the rational function becomes zero. In this case, the denominator is x^n - 5. To have no vertical asymptotes, the denominator must never equal zero.
If we set x^n - 5 = 0 and solve for x, we get:
x^n = 5
Taking the n-th root of both sides, we get:
x = 5^(1/n)
For there to be no vertical asymptotes, the value of x = 5^(1/n) should not be a real number. However, since we are dealing with real numbers, we can conclude that there will always be a value of x that makes the denominator zero, and thus, the function will always have vertical asymptotes.
Example:
Let's take a = 1 and n = 2. The function becomes:
f(x) = x^2 / x^2 - 5
When we set the denominator x^2 - 5 to zero and solve for x, we get:
x^2 = 5
x = √5 and x = -√5
Therefore, the function will have vertical asymptotes at x = √5 and x = -√5.
b) A horizontal asymptote occurs when the degree of the numerator is less than or equal to the degree of the denominator. In this case, the degree of the numerator is 2 and the degree of the denominator is n.
If 2 ≤ n, then the function will have a horizontal asymptote at y = 0.
Example:
Let's take a = 1 and n = 3. The function becomes:
f(x) = x^2 / x^3 - 5
Here, the degree of the numerator (2) is less than the degree of the denominator (3). Therefore, the function will have a horizontal asymptote at y = 0.
a) To determine if there are any values of a and n for which the function f(x) = ax^2 / (x^n - 5) will have no vertical asymptotes, we need to consider the denominator (x^n - 5).
A vertical asymptote occurs at x = a if and only if the denominator of the function approaches zero as x approaches a. In this case, we want to find values of a and n such that the denominator does not approach zero for any real value of x.
To do this, we need to find the solutions to the equation x^n - 5 = 0. Setting the denominator equal to zero, we have:
x^n - 5 = 0
To solve this equation, we can isolate x by adding 5 to both sides:
x^n = 5
Then, we can take the nth root of both sides:
x = (5)^(1/n)
Now, let's analyze this expression. The value of (5)^(1/n) will always be positive for any positive value of n. This means that the denominator x^n - 5 will never equal zero for any real value of x.
Therefore, there are no values of a and n for which the function f(x) = ax^2 / (x^n - 5) will have vertical asymptotes.
b) To determine if there are any values of a and n for which the function f(x) = ax^2 / (x^n - 5) will have a horizontal asymptote, we need to examine the behavior of the function as x approaches positive or negative infinity.
A horizontal asymptote occurs at y = b if and only if the function approaches the value b as x approaches positive or negative infinity.
In this case, since both the numerator (ax^2) and the denominator (x^n - 5) are polynomials, their behavior as x approaches positive or negative infinity can be determined by the degree of the polynomial in the numerator and the denominator.
If the degree of the polynomial in the denominator is greater than the degree of the polynomial in the numerator, then the function will have a horizontal asymptote at y = 0.
However, if the degree of the polynomial in the numerator is equal to or greater than the degree of the polynomial in the denominator, then the function will not have a horizontal asymptote.
In our case, the degree of the polynomial in the numerator is 2 (since it is ax^2), and the degree of the polynomial in the denominator is n (since it is x^n - 5). If n is equal to or higher than 2, then the function will not have a horizontal asymptote.
For example, let's consider the case when n = 2. The function becomes:
f(x) = ax^2 / (x^2 - 5)
Since the degree of the polynomial in the numerator is equal to the degree of the polynomial in the denominator, the function does not have a horizontal asymptote.
Therefore, there are no values of a and n for which the function f(x) = ax^2 / (x^n - 5) will have a horizontal asymptote.