The graph of the function f(x)=(x^-9)/2x^2-5x-3 has a vertical asymptote a x=a and a horizontal asymptote at y=b. What are the values of the constants a and b?
A. a=-3 b=2
B. a=-1/2 b=1/2
C. a=1/2 b=1/2
D. a=3 b=2
B???
yes.
To find the values of the constants a and b, we need to analyze the behavior of the function f(x) as x approaches positive or negative infinity.
First, let's determine the vertical asymptote, which occurs when the denominator of the fraction f(x) approaches zero.
The denominator is 2x^2 - 5x - 3. We can find the solutions to this quadratic equation by factoring or using the quadratic formula.
The factored form of 2x^2 - 5x - 3 is (2x + 1)(x - 3).
Setting each factor equal to zero, we get 2x + 1 = 0 or x - 3 = 0.
Solving these equations, we find x = -1/2 or x = 3.
Therefore, the vertical asymptotes are x = -1/2 and x = 3.
Now, let's consider the horizontal asymptote, which describes the behavior of the function as x approaches positive or negative infinity.
To find the horizontal asymptote, we look at the degree of the numerator and denominator. The degree of the numerator is -9, and the degree of the denominator is 2.
Since the numerator's degree is less than the denominator's degree, the horizontal asymptote will be at y = 0.
Comparing the options given, we can see that none of them match the solution of a= -1/2 and b= 0.
Therefore, the correct answer is none of the choices provided.