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Chapter 4: 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 because there are no real numbers that you can substitute into the function to make it undefined.

Step by step solution

01

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.
02

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.
03

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.

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Most popular questions from this chapter

The table shows the percents \(P\) of women in different age groups (in years) who have been married at least once. (Source: U.S. Census Bureau) $$\begin{array}{|c|c|}\hline \text { Age group } & \text { Percent, } P\\\\\hline 18-24 & 14.6 \\\25-29 & 49.0 \\\30-34 & 70.3 \\\35-39 & 79.9 \\\40-44 & 85.0 \\\45-49 & 87.0 \\\50-54 & 89.5 \\\55-59 & 91.1 \\\\\hline\end{array}$$ (a) Use the regression feature of a graphing utility to find a logistic model for the data. Let \(x\) represent the midpoint of the age group. (b) Use the graphing utility to graph the model with the original data. How closely does the model represent the data?

A logarithmic equation can have at most one extraneous solution.

Determine whether the statement is true or false. Justify your answer. The exponential model \(y=a e^{b x}\) represents a growth model when \(b>0.\)

Use the regression feature of a graphing utility to find an exponential model \(y=a b^{x}\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(0,4.0),(2,6.9),(4,18.0),(6,32.3),(8,59.1),(10,118.5)$$

Use a graphing utility to create a scatter plot of the data. Decide whether the data could best be modeled by a linear model, an exponential model, or a logarithmic model. $$(1,11.0),(1.5,9.6),(2,8.2),(4,4.5),(6,2.5),(8,1.4)$$
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