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Chapter 8: Problem 103

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I modeled California's population growth with a geometric sequence, so my model is an exponential function whose domain is the set of natural numbers.

Short Answer

Expert verified
The statement makes sense as geometric sequences can be represented as exponential functions and the domain described (set of natural numbers) suits the real-world application described.

Step by step solution

01

Concept of Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the ratio. This ratio stays the same throughout the sequence.
02

Concept of Exponential Function

An exponential function is a mathematical function of the following form: \( f(x) = a \cdot r^x \) where \( a \) and \( r \) are real numbers, and \( r \) is positive and does not equal 1. The variable \( x \) is the exponent, while \( a \) is the coefficient.
03

Relationship Between Geometric Sequence and Exponential Function

Geometric sequences can, indeed, be modeled by exponential functions. Concrete, a geometric sequence with the formula \( a_n = a \cdot r^{(n-1)} \) is very similar to an exponential function of form \( f(x) = a \cdot r^x \). In the geometric sequence, \( n \) is restricted to integers as it represents the position in the sequence.
04

Concept of Domain

In mathematics, the domain of a function is the set of all possible input values (often referred to as 'x-values') to which the function is defined.
05

Validation of the Statement

The statement about modelling California's population growth as a geometric sequence, and hence as an exponential function whose domain is the set of natural numbers, makes sense. That's because, in a real world scenario like population growth, \( n \) would be equivalent of time (in years, for example) and it would make sense for it to take on only natural numbers. Thus, the given domain matches the scenario.

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