To determine whether the function \( g(x) \) is linear or exponential, we can analyze the growth of the values given for \( g(x) \).
Step 1: Check for Linearity
A linear function has a constant rate of change (slope). We can calculate the differences between consecutive values of \( g(x) \):
\[ \begin{align*} g(0) - g(-1) &= 3 - \frac{3}{2} = \frac{3}{2} = 1.5 \ g(1) - g(0) &= 6 - 3 = 3 \ g(2) - g(1) &= 12 - 6 = 6 \ g(3) - g(2) &= 24 - 12 = 12 \ \end{align*} \]
The differences are:
- \( 1.5 \) (from -1 to 0)
- \( 3 \) (from 0 to 1)
- \( 6 \) (from 1 to 2)
- \( 12 \) (from 2 to 3)
Since the differences (1.5, 3, 6, 12) are not constant, \( g(x) \) is not linear.
Step 2: Check for Exponential Growth
An exponential function has a constant ratio between consecutive values. We can calculate the ratios of consecutive values of \( g(x) \):
\[ \begin{align*} \frac{g(0)}{g(-1)} &= \frac{3}{\frac{3}{2}} = 2 \ \frac{g(1)}{g(0)} &= \frac{6}{3} = 2 \ \frac{g(2)}{g(1)} &= \frac{12}{6} = 2 \ \frac{g(3)}{g(2)} &= \frac{24}{12} = 2 \ \end{align*} \]
The ratios are all equal to \( 2 \), which indicates that \( g(x) \) exhibits exponential growth.
Step 3: Write the Exponential Equation
Since \( g(x) \) can be expressed in exponential form, we can use the following general equation:
\[ g(x) = a(b^x) \]
Given the ratios, we can identify \( b = 2 \). Let's find \( a \) using one of the known points. We can use \( g(0) = 3 \):
\[ g(0) = a(2^0) = a(1) = a \implies a = 3 \]
Thus, we can write the function as:
\[ g(x) = 3(2^x) \]
Final Result
The function \( g(x) \) is exponential, and the equation in the desired form is:
\[ g(x) = 3(2^x) \]
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