Problem 61 - Algebra and Trigonometry Real Mathematics, Real People

Question

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.

Step-by-Step Solution

Answer

The statement is false. The domain of a logistic growth function can be the set of all real numbers because there are no real numbers that you can substitute into the function to make it undefined.

Step 1: Understand a Logistic Function

Logistic functions are used in a variety of fields, including statistics, ecology, and economics, to express growth under constraints. When the x represents time, the logistic function yields a graph which starts with a phase of exponential growth before slowing and flattening out - ideal for representing situations where growth is initially rapid but then becomes constrained. It is typically expressed in the form \(y = \frac{c}{1 + ae^{-bx}}\), where c, a, and b are constants.

Step 2: Reflect on the Domain and Real Numbers

The domain of a function includes all the possible values that the variable x can be, such that a y-value (output) can be determined by the function. The set of all real numbers includes all rational and irrational numbers, which essentially means any number on the number line.

Step 3: Evaluate if the Domain of a Logistic Function can be the set of all Real Numbers

We need to determine whether there's any real number we can put in for x in the logistic growth function \(y = \frac{c}{1 + ae^{-bx}}\) that would make this function undefined. Looking closely, we see that the denominator of the fraction can never be zero because there's a '+1' added to \(ae^{-bx}\). Since dividing by zero is the only operation that would make the function undefined, we can safely say that logistic growth functions are defined for all real numbers.