The following rational function describes concentration in blood of a certain medicine taken once depending on time, find
a. the horizontal or oblique asymptote(s), if any,
b. the vertical asymptote(s), if any,
c. describe their possible meanings.
f(x)=4/x+2
To find the asymptotes of the rational function f(x) = 4/(x + 2), we need to consider the behavior of the function as x approaches certain values.
a. Horizontal or Oblique Asymptote(s):
To find the horizontal or oblique asymptotes, we need to compare the degrees of the numerator and denominator terms.
In this case, the degree of the numerator is 0 (constant term 4), and the degree of the denominator is 1 (linear term x). Since the degree of the numerator is smaller than the degree of the denominator, there is a horizontal asymptote at y = 0.
b. Vertical Asymptote(s):
To find the vertical asymptotes, we need to determine any x-values that make the denominator zero.
In this case, we find that x + 2 = 0 when x = -2. Therefore, there is a vertical asymptote at x = -2.
c. Possible Meanings:
The horizontal asymptote at y = 0 indicates that as time progresses, the concentration in the blood of the medicine taken once approaches zero. This suggests that the medicine is being eliminated from the body.
The vertical asymptote at x = -2 suggests that there is a restriction on the possible values of time. It could mean that the medicine has a limited duration of action or that it is metabolized and eliminated from the body after a certain amount of time.