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Chapter 3: Problem 61

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
The statement is false. The domain of a logistic growth function can be the set of all real numbers.

Step by step solution

01

Understanding the logistic function

The logistic growth function is given by \(f(x) = \frac{L}{1 + e^{-k(x - x_0)}}\), where \(L\), \(k\), and \(x_0\) are constants, and \(e\) is the base of natural logarithm. The function exists for any value of \(x\) because for any real number \(x\), the value of Gaussian function \(e^{-k(x - x_0)}\) is always a positive real number, and therefore the function is defined.
02

Determine the domain

The domain of function is all the valid input values that the function can take. For the logistic function, the function is defined and real for any real value of \(x\). So the domain of a logistic growth function can indeed be the set of all real numbers.
03

Final conclusion

Therefore, the statement 'The domain of a logistic growth function cannot be the set of all real numbers.' is false.

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