Question

Consider the various types of functions that can be used for mathematical models, which types of functions could be used to describe a situation in which the number of individuals in an endangered population (the dependent variable) becomes asymptotically close to reaching zero but never actually becomes extinct ? Justify your choice of functions

Answer 1

• Rational function: y = a/x, ,x ≥ 0

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• Exponential function: y = a⁻ˣ

Step-by-step explanation

We have to select functions that have a horizontal asymptote at y = 0, so the graph of these functions must have the form,

It must be decreasing because the dependent variable represents the number of individuals in a population, so it must be a positive number.

One of the functions that have this behavior is a rational function y = a/x. As x goes to infinity, y approaches 0. This happens in the other direction, as x goes to negative infinity, y approaches 0. In this case, we would only consider the values for x ≥ 0.

Another function that has this kind of asymptotic behavior is an exponential function where the exponent is negative: y = a⁻ˣ. When x = 0, y = 1, and then it decreases approaching the x-axis but never touching it.

Hence, the two types of functions that have this behavior are y = a/x and y = a⁻ˣ, where a is a constant.