Solve each problem. Suppose that the population of a species...

Solve each problem. Suppose that the population of a species of fish in thousands is modeled by $$f(x)=\frac{x+10}{0.5 x^{2}+1}$$ where $x \geq 0$ is in years. (a) Graph $f$ in the window $[0,12]$ by $[0,12] .$ What is the equation of the horizontal asymptote? (b) Determine the initial population. (c) What happens to the population after many years? (d) Interpret the horizontal asymptote.

Solve each problem.
Suppose that the population of a species of fish in thousands is modeled by
$$f(x)=\frac{x+10}{0.5 x^{2}+1}$$
where $x \geq 0$ is in years.
(a) Graph $f$ in the window $[0,12]$ by $[0,12] .$ What is the equation of the horizontal asymptote?
(b) Determine the initial population.
(c) What happens to the population after many years?
(d) Interpret the horizontal asymptote.

Key Concepts

Interpretation in Population Dynamics

In the context of population models, a horizontal asymptote represents the value that the population approaches as time goes to infinity. Even if the initial population is high, external factors causing a decline may drive the population toward the asymptotic level, here indicating eventual elimination or significant reduction of the species if the dynamics continue unchanged.

Long-Term Behavior and Limits

The concept of limits is used to study the behavior of a function as the input grows without bound. For rational functions, if the degree of the denominator is higher than that of the numerator, the function’s value approaches 0 as x increases. This reflects how the system behaves in the long run.

Rational Functions and Horizontal Asymptotes

A rational function is a ratio of two polynomials. When analyzing such functions, the degrees of the numerator and denominator are crucial. If the degree of the numerator is lower than the denominator, the horizontal asymptote is y = 0. This principle helps determine the end behavior of the function as x becomes very large.

Initial Value Analysis

The initial value of a function gives the output when x is zero, which in applied models like population studies represents the population at the initial time. Evaluating the function at x = 0 provides insight into the starting conditions of the system being modeled.

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