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Chapter 3: Problem 79

Determine whether the statement is true or false. Justify your answer. The domain of a logistic growth function cannot be the set of real numbers.

Short Answer

Expert verified
False. The domain of a logistic growth function can be the set of all real numbers.

Step by step solution

01

Understanding Logistic Growth Function

A logistic growth function, also known as a sigmoid function, is a common 'S' shape (sigmoid curve) function. It can be written in the form \(f(x) = \frac{C}{1 + ae^{-bx}} \), where 'e' is Euler's number (approx. 2.71828), 'C' represents the limit to growth (called carrying capacity), 'a' is any real number and 'b' is the growth rate. 'x' is the independent variable in this formula.
02

Understanding the domain of a function

The domain of a function is the complete set of possible values of the independent variable. In simpler words, it is the set of all possible x-values which will make the function 'work', and will output real y-values.
03

Determine the Domain of Logistic Function

Looking at the logistic function definition, there is no constraint on 'x' that would prevent it from being any real number. The function is well-defined for all real values of 'x'. The expression \(1 + ae^{-bx} \) is not violated for any real numbers. Therefore, the domain of a logistic function is the set of all real numbers, R.

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Most popular questions from this chapter

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log 4 x-\log (12+\sqrt{x})=2$$

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log 8 x-\log (1+\sqrt{x})=2$$

You are investing \(P\) dollars at an annual interest rate of \(r,\) compounded continuously, for \(t\) years. Which of the following would result in the highest value of the investment? Explain your reasoning. (a) Double the amount you invest. (b) Double your interest rate. (c) Double the number of years.

The numbers \(y\) of freestanding ambulatory care surgery centers in the United States from 2000 through 2007 can be modeled by \(y=2875+\frac{2635.11}{1+14.215 e^{-0.8038 t}}, \quad 0 \leq t \leq 7\) where \(t\) represents the year, with \(t=0\) corresponding to 2000 . (Source: Verispan) (a) Use a graphing utility to graph the model. (b) Use the trace feature of the graphing utility to estimate the year in which the number of surgery centers exceeded \(3600 .\)

(a) solve for \(P\) and (b) solve for \(t\). $$A=P e^{r t}$$
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