Problem 103 - College Algebra

Question

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.

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.

Step 5: 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.