The following rational function describes blood concentration of a certain drug taken via IV over time, find
a. the horizontal or oblique asymptotes
b. the vertical asymptote
c. describe their possible meanings
f(x)= x+1/x
f(x)= x+1/x
no horizontal asymptote
vertical: x=0
slant asymptote: y = x
Can't give any possible meaning, since the function is poorly defined
mathematical meaning: as x increases so does f(x)
but in terms of blood concentration, it makes no physical sense for small values of x
see:
http://www.wolframalpha.com/input/?i=f%28x%29%3D+x%2B1%2Fx+for+x+from+0+to+10
You might want to look at the first 7 of the Related Questions below, all a variation of yours.
Most are unanswered, probably due to their ambiguity. Steve tried a few of them but had the same problem
To find the asymptotes of the given rational function f(x) = (x + 1)/x, we need to analyze the behavior of the function as x approaches positive infinity, negative infinity, and at vertical asymptotes (if any).
a. Horizontal or Oblique Asymptotes:
To find horizontal or oblique asymptotes, we need to compare the degrees of the numerator and denominator. In this case, the degree of the numerator is 1, and the degree of the denominator is also 1. Since the degrees are the same, there is no horizontal asymptote.
b. Vertical Asymptote:
To find the vertical asymptote, we need to identify the values of x for which the denominator becomes zero. In this case, the denominator x becomes zero when x = 0.
c. Possible Meanings:
- Horizontal or Oblique Asymptotes: Since there are no horizontal or oblique asymptotes in this case, it suggests that the blood concentration of the drug taken via IV does not tend to approach a constant value or a specific linear relationship as time goes on.
- Vertical Asymptote: The vertical asymptote at x = 0 indicates that the concentration of the drug cannot be measured at time t = 0 or at any negative time. It suggests that the drug needs time to enter the bloodstream and achieve a measurable concentration.
In summary:
a. There are no horizontal or oblique asymptotes.
b. The vertical asymptote is x = 0.
c. The absence of horizontal or oblique asymptotes suggests no long-term convergence or relationship, while the vertical asymptote indicates that the drug concentration cannot be measured right at the beginning (t = 0).
To find the asymptotes of a rational function, we need to analyze its behavior as x approaches infinity (horizontal or oblique asymptotes) and as x approaches certain values (vertical asymptotes).
a. To find the horizontal or oblique asymptotes, we need to examine the degree of the numerator and the denominator of the rational function.
In this case, the degree of the numerator is 1 (since there's an x term) and the degree of the denominator is also 1 (since there's an x term in the denominator as well). The degrees are equal, so we can look at the ratio of the leading coefficients.
The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is also 1. Therefore, the ratio of the leading coefficients is 1/1 = 1.
When the ratio of the leading coefficients is a finite number (not zero), the rational function has a horizontal asymptote at y = a/b, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator. In this case, the horizontal asymptote is y = 1/1 = 1.
b. To find the vertical asymptote(s), we check for values of x that make the denominator equal to zero.
In this function, the denominator is x. Setting x = 0, we have a vertical asymptote at x = 0.
c. The possible meanings of the asymptotes in the context of blood concentration of a drug are as follows:
- The horizontal asymptote at y = 1 indicates that as time goes on, the blood concentration of the drug will approach a steady state or equilibrium level. Regardless of the value of x, the drug concentration will tend to stabilize at a level of 1.
- The vertical asymptote at x = 0 suggests that the drug may not be administered at time zero or that there may be some restrictions or limitations in giving the drug at that specific time.
Note: Without further context or additional information, this interpretation is a general description and may not reflect the specific meaning in a given situation.