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Chapter 3: Problem 62

Determine whether the statement is true or false. Justify your answer. A logistic growth function will always have an \(x\) -intercept.

Short Answer

Expert verified
The statement is false. A logistic growth function does not have an x-intercept.

Step by step solution

01

Understand the nature of a logistic growth function

A logistic function is typically used to model population growth under the assumption that resources are limited. The function is defined by \(f(x) = \frac{L}{1+e^{-k(x-x_0)}}\), where L represents the maximum y-value (also called carrying capacity), k is the growth rate, and \(x_0\) is the x-value of the sigmoid's midpoint.
02

Determine when the function equals zero

To find the x-intercepts, we need to find the x values such that \(f(x) = 0\). Setting \(f(x) = 0\) yields \n0 = \frac{L}{1+e^{-k(x-x_0)}}. However, there is no real solution for \(x\) in this equation. This is because no matter what the value of \(x\), the logistic function always gives an output greater than 0.
03

Conclusion of the analysis

Therefore, a logistic growth function does not have an x-intercept in real numbers.

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