- Which of the following is false for f(x) = (5x^3-5x^2-10x)/(2x^5-2x)
a) The x-axis is an asymptote of f(x)
b) x = -1 is not an asymptote of f(x).
c) x = 1 is an asymptote of f(x).
d) The y-axis is an asymptote of f(x).
Thanks
d is false. The others are all true. See
http://www.wolframalpha.com/input/?i=(5x%5E3-5x%5E2-10x)%2F(2x%5E5-2x)
To determine which of the statements is false for the function f(x) = (5x^3 - 5x^2 - 10x)/(2x^5 - 2x), let's analyze each statement individually:
a) The x-axis is an asymptote of f(x):
To find the x-axis asymptote, we need to determine if the function approaches a particular value as x approaches positive or negative infinity.
Let's simplify the function:
f(x) = (5x^3 - 5x^2 - 10x)/(2x^5 - 2x)
= (5x(x^2 - x - 2))/(2x(x^4 - 1))
= (5(x^2 - x - 2))/(2(x^4 - 1))
As x approaches infinity, the expression (x^2 - x - 2) in the numerator will dominate the expression (x^4 - 1) in the denominator. Thus, the function approaches positive infinity, but not zero (which would be necessary for the x-axis to be an asymptote). Therefore, statement a) is false.
b) x = -1 is not an asymptote of f(x):
To evaluate the behavior of the function near x = -1, let's substitute x = -1 into the function:
f(-1) = (5(-1)^3 - 5(-1)^2 - 10(-1))/(2(-1)^5 - 2(-1))
= (-5 - 5 + 10)/(-2 + 2)
= 0/0
The result is an indeterminate form of 0/0, indicating that further analysis is required. To determine if x = -1 is an asymptote, we can factorize the numerator and denominator and then check if the factor (x - (-1)) = (x + 1) cancels out.
The numerator factorizes as:
5x(x^2 - x - 2) = 5x(x - 2)(x + 1)
The denominator factorizes as:
2x(x^4 - 1) = 2x(x^2 - 1)(x^2 + 1) = 2x(x - 1)(x + 1)(x^2 + 1)
By canceling out the common factors of (x + 1) in the numerator and denominator, we get:
f(x) = (5x)/(2(x^2 - 1))
Thus, x = -1 is indeed not an asymptote, since it cancels out in the simplified function. Therefore, statement b) is true.
c) x = 1 is an asymptote of f(x):
To evaluate the behavior of the function near x = 1, let's substitute x = 1 into the function:
f(1) = (5(1)^3 - 5(1)^2 - 10(1))/(2(1)^5 - 2(1))
= (5 - 5 - 10)/(2 - 2)
= -10/0
The result is an indeterminate form of -10/0, indicating that further analysis is required. To determine if x = 1 is an asymptote, we can factorize the numerator and denominator and then check if the factor (x - 1) cancels out.
The numerator factorizes as:
5x(x^2 - x - 2) = 5x(x - 2)(x + 1)
The denominator factorizes as:
2x(x^4 - 1) = 2x(x^2 - 1)(x^2 + 1) = 2x(x - 1)(x + 1)(x^2 + 1)
By canceling out the common factor of (x - 1) in the numerator and denominator, we get:
f(x) = (5x(x - 2))/(2(x + 1)(x^2 + 1))
Thus, x = 1 is not an asymptote, since it cancels out in the simplified function. Therefore, statement c) is false.
d) The y-axis is an asymptote of f(x):
To find the y-axis asymptote, we need to determine if the function approaches a particular value as x approaches zero or infinity.
As x approaches zero, both the numerator and denominator approach zero. However, the highest power of x in the denominator (x^5) is greater than the highest power of x in the numerator (x^3). Therefore, as x approaches zero, the function approaches zero, not infinity. Thus, the y-axis is not an asymptote, and statement d) is true.
In summary, the false statement among the options is c) x = 1 is an asymptote of f(x).
To determine which statements are true or false, we need to analyze the behavior of the function f(x) = (5x^3 - 5x^2 - 10x) / (2x^5 - 2x).
To find the vertical asymptotes, we need to identify the values of x where the denominator of the function equals zero. Set the denominator equal to zero and solve for x:
2x^5 - 2x = 0
Factor out the common term 2x:
2x(x^4 - 1) = 0
Now, we have two factors: 2x = 0 and (x^4 - 1) = 0
Solving the first equation, we find that x = 0. Therefore, the y-axis (x = 0) can potentially be an asymptote.
To solve the second equation, we need to factor it further:
(x^2 + 1)(x^2 - 1) = 0
Now, solve each factor separately:
x^2 + 1 = 0 has no real solutions.
x^2 - 1 = 0 gives us x = -1 and x = 1.
So, we have three potential vertical asymptotes: x = 0, x = -1, and x = 1.
Now, let's evaluate each statement:
a) The x-axis is an asymptote of f(x). - This statement is FALSE. The x-axis (y = 0) is not among the potential vertical asymptotes we found.
b) x = -1 is not an asymptote of f(x). - This statement is TRUE. x = -1 is not among the potential vertical asymptotes. However, to be completely sure, we need to check if it is a hole or removable singularity, which cannot be determined without additional information.
c) x = 1 is an asymptote of f(x). - This statement is TRUE. x = 1 is one of the potential vertical asymptotes we found.
d) The y-axis is an asymptote of f(x). - This statement is TRUE. The y-axis (x = 0) is among the potential vertical asymptotes.
So, the correct answer is:
- Statement a) is false,
- Statement b) is true,
- Statement c) is true, and
- Statement d) is true.