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Chapter 3: Problem 80

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
False. A logistic growth function will not have an \(x\)-intercept.

Step by step solution

01

Understanding the Logistic Function

The logistic function growth starts slow, increases rapidly, and then slows down again, making an S-shaped curve. It is bound above by a horizontal asymptote which is the maximum value \(L\). However, it doesn't have a lower bound.
02

Evaluating x-intercept

The \(x\)-intercept is defined as the point where the curve crosses the \(x\)-axis. This happens when \(f(x) = 0\). Setting the logistic growth function to zero gives \(0 = \frac{L}{1 + e^{-k(x-x0)}}\). Because \(L>0\), this equation has no solution. The denominator is always positive due to the properties of exponential function, and the fraction will be zero only if \(L = 0\), which contradicts the definition of a logistic growth function. Hence, a logistic growth function can't have an \(x\)-intercept.
03

Conclusion

Therefore, the statement 'A logistic growth function will always have an \(x\)-intercept' is false. The logistic growth function will approach the \(x\)-axis but never cross it, so it does not have an \(x\)-intercept.

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