Steve,
The following rational function describes concentration in blood of a certain medicine taken once depending on time, find: A. the horizontal asymptote B. the vertical asymptote C. describe their possible meaning ( I only need help with C). f(x)=x/x^2-100
Sharon
http://www.wolframalpha.com/input/?i=x%2F%28x^2-100%29
as you can see from the graph, the concentration appears to be negative until x=10. Then it explodes and then drops down.
I cannot think what the vertical asymptote can represent, unless there's some delayed-release capsule, and the asymptote is just an approximation to the sudden release of medicine. I think that's a bigus function.
Thank you
To find the meaning of the horizontal and vertical asymptotes, we first need to understand what they represent in the context of a rational function.
A rational function is defined as a ratio of two polynomials. In this case, the rational function is given as f(x) = x / (x^2 - 100). The numerator is x, and the denominator is x^2 - 100.
Now let's explain the meaning of the horizontal and vertical asymptotes in the context of this rational function:
A. Horizontal Asymptote:
In a rational function, the horizontal asymptote represents the behavior of the function as x approaches positive or negative infinity. To find the horizontal asymptote, we need to examine the degrees of the numerator and denominator.
In this case, the degree of the numerator is 1 (degree of x), and the degree of the denominator is 2 (degree of x^2). When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote for the function is at y = 0.
So, the horizontal asymptote for the given function is y = 0. This means that as time goes on (as x approaches positive or negative infinity), the concentration in the blood of the medicine taken once will approach zero.
B. Vertical Asymptote:
A vertical asymptote represents the values of x for which the function approaches positive or negative infinity. To find the vertical asymptote, we need to determine the values of x that make the denominator of the rational function equal to zero.
In this case, the denominator is x^2 - 100. To find the values of x that make the denominator zero, we set x^2 - 100 = 0 and solve for x.
x^2 - 100 = 0
This equation can be factored as (x - 10)(x + 10) = 0.
Setting each factor equal to zero, we get:
x - 10 = 0 or x + 10 = 0
Solving for x, we find two vertical asymptotes:
x = 10 and x = -10
These vertical asymptotes at x = 10 and x = -10 indicate that the concentration in the blood of the medicine taken once will approach infinity as x approaches these values.
C. Possible Meaning:
The possible meaning of the horizontal asymptote at y = 0 is that as time goes on, the concentration of the medicine in the blood will eventually approach zero. This suggests that the medicine is being eliminated or metabolized by the body over time.
The vertical asymptotes at x = 10 and x = -10 indicate that there may be certain time points where the concentration of the medicine spikes or increases significantly. This could be due to factors like the medicine's absorption or metabolism being affected at those specific time points.
It's important to note that these interpretations are based on the mathematical analysis of the given rational function and may not necessarily reflect real-world scenarios. The actual meaning and interpretation of the vertical and horizontal asymptotes would depend on the specific context and characteristics of the medicine.