The values of two functions, f and g, are given in a table....

The values of two functions, $f$ and $g$, are given in a table. One, both, or neither of them may be exponential. Decide which, if any, are exponential, and give the exponential models for those that are. HINT [See Example 1.] $$ \begin{array}{|c|c|c|c|c|c|} \hline \boldsymbol{x} & -2 & -1 & 0 & 1 & 2 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 100 & 200 & 400 & 600 & 800 \\ \hline \boldsymbol{g}(\boldsymbol{x}) & 100 & 20 & 4 & 0.8 & 0.16 \\ \hline \end{array} $$

The values of two functions, $f$ and $g$, are given in a table. One, both, or neither of them may be exponential. Decide which, if any, are exponential, and give the exponential models for those that are. HINT [See Example 1.]
$$
\begin{array}{|c|c|c|c|c|c|}
\hline \boldsymbol{x} & -2 & -1 & 0 & 1 & 2 \\
\hline \boldsymbol{f}(\boldsymbol{x}) & 100 & 200 & 400 & 600 & 800 \\
\hline \boldsymbol{g}(\boldsymbol{x}) & 100 & 20 & 4 & 0.8 & 0.16 \\
\hline
\end{array}
$$

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Key Concepts

Exponential Function

An exponential function is characterized by a constant percentage or multiplicative change, rather than a constant additive change. It can be modeled using the formula y = ab^x, where 'a' is the initial value and 'b' is the constant base that indicates the rate of growth (if b > 1) or decay (if 0 < b < 1). This form captures the idea that the function values change by the same proportion for equal increments in x.

Common Ratio

The common ratio is the fixed factor by which consecutive values of an exponential function differ. When analyzing a table of function outputs, if the ratio of any term to its previous term is constant, the function is exponential. This means that multiplying one term by this constant yields the next term, confirming the behavior predicted by an exponential model.

Constructing an Exponential Model

To construct an exponential model from data, one typically uses the formula y = ab^x. Here, the initial value 'a' is often determined by the output at a known x-value (commonly x = 0), and the base 'b' is found by taking the ratio of consecutive outputs. This approach allows for the verification and formulation of an exponential equation that fits the observed values.

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