Question

which exponential function is represented by the values in the table? \\(\

$$\begin{array}{|c|c|}\\hline x&f(x)\\\\\\hline -2&\\frac{3}{4}\\\\\\hline -1&\\frac{3}{2}\\\\\\hline 0&3\\\\\\hline 1&6\\\\\\hline 2&12\\\\\\hline \\end{array}$$

\\) options: \\(f(x) = 3(3^x)\\), \\(f(x) = 2(2^x)\\), \\(f(x) = 3(2^x)\\), \\(f(x) = 2(3^x)\\)

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( f(x) = a(b^x) \), where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the common ratio.

Step2: Find the initial value \( a \)

When \( x = 0 \), from the table, \( f(0) = 3 \). Substituting \( x = 0 \) into \( f(x) = a(b^x) \), we get \( f(0)=a(b^0)=a(1)=a \). So \( a = 3 \).

Step3: Find the common ratio \( b \)

Let's check the ratio between consecutive \( f(x) \) values. For \( x = 0 \) to \( x = 1 \): \( \frac{f(1)}{f(0)}=\frac{6}{3} = 2 \). For \( x = 1 \) to \( x = 2 \): \( \frac{f(2)}{f(1)}=\frac{12}{6}=2 \). So the common ratio \( b = 2 \).

Step4: Determine the function

Substituting \( a = 3 \) and \( b = 2 \) into the exponential function form, we get \( f(x)=3(2^x) \).

Answer:

\( f(x) = 3(2^x) \) (corresponding to the option \( f(x) = 3(2^x) \))