To determine the type of function that describes [tex]\( g(x) \)[/tex], we can analyze the given table of values.
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & -3 & -1 & 1 & 3 & 5 & 7 & 9 \\
\hline
g(x) & 0.375 & 0.75 & 1.5 & 3 & 6 & 12 & 24 \\
\hline
\end{array}
\][/tex]
Step-by-step solution:
1. Identify Rate of Change: Compare the ratio of consecutive [tex]\( g(x) \)[/tex] values. For an exponential function, the ratio of consecutive [tex]\( g(x) \)[/tex] values should be constant.
2. Calculate Ratios:
[tex]\[
\frac{g(-1)}{g(-3)} = \frac{0.75}{0.375} = 2.0
\][/tex]
[tex]\[
\frac{g(1)}{g(-1)} = \frac{1.5}{0.75} = 2.0
\][/tex]
[tex]\[
\frac{g(3)}{g(1)} = \frac{3}{1.5} = 2.0
\][/tex]
[tex]\[
\frac{g(5)}{g(3)} = \frac{6}{3} = 2.0
\][/tex]
[tex]\[
\frac{g(7)}{g(5)} = \frac{12}{6} = 2.0
\][/tex]
[tex]\[
\frac{g(9)}{g(7)} = \frac{24}{12} = 2.0
\][/tex]
3. Check Consistency: All these ratios are equal to [tex]\( 2.0 \)[/tex]. This consistent ratio confirms that the function describing [tex]\( g(x) \)[/tex] is exponential.
4. Conclusion: Since each ratio of consecutive [tex]\( g(x) \)[/tex] values is constant (2.0 in this case), [tex]\( g(x) \)[/tex] can be described by an exponential function.
So, the function [tex]\( g(x) \)[/tex] is Exponential.
