The following rational function describes concentration in blood of a certain medicine taken once depending on time, find:
A) the horizontal or oblique asymptote
B) the vertical asymptote, if any
C) describe their meanings
f(x)=x/x^2-100
Since x has degree less than x^2-100, the horizontal asymptote is y=0
since x^2-100 = 0 at x = ±10, and x≠0, those are the vertical asymptotes
I find the function unrealistic. It implies negative concentrations, going off the charts as time increases. I suspect a function like x/(x^2+100) would be more like it. It starts with zero concentration, builds up as the pill is digested, and then tapers off at the medicine is metabolized.
To find the horizontal or oblique asymptote of the given rational function f(x), we need to examine the behavior of the function as x approaches positive or negative infinity.
A) Horizontal or Oblique Asymptote:
1. Determine the degree of the numerator and denominator of the rational function.
- The degree of the numerator is 1 (since the highest power of x in the numerator is 1).
- The degree of the denominator is also 1 (since the highest power of x in the denominator is 2).
2. Compare the degrees of the numerator and denominator:
- Since the degree of the numerator is less than the degree of the denominator, the equation does not have a horizontal asymptote at y = 0.
3. Determine if there is an oblique asymptote:
- To find if an oblique asymptote exists, divide the function's numerator by its denominator using polynomial long division or synthetic division.
Dividing x by x^2 - 100 gives:
1
----------
x^2 - 100 | x
- (x^2 - 100)
-------------
100
The result is 0 with a remainder of 100.
Therefore, the rational function does not have an oblique asymptote as the division resulted in a remainder of 100.
B) Vertical Asymptote:
1. Find the values of x that make the denominator of the rational function equal to zero.
- In this case, the denominator is x^2 - 100.
Setting x^2 - 100 = 0 and solving for x:
x^2 - 100 = 0
(x - 10)(x + 10) = 0
From here, we can see that x = -10 and x = 10 are the values that make the denominator equal to zero. Thus, the vertical asymptotes are x = -10 and x = 10.
C) Meaning of the Asymptotes:
1. Horizontal or Oblique Asymptote:
- Since the function does not have a horizontal asymptote at y = 0 or an oblique asymptote, it implies that the concentration in blood of the medicine taken once does not tend to approach a specific value as time goes to infinity.
2. Vertical Asymptote:
- The vertical asymptotes at x = -10 and x = 10 indicate that the concentration in blood may become unbounded or undefined at these points. It suggests that there may be conditions or factors where the medication is not effective or poses an adverse effect on the body when certain amounts are present in the blood.