Questions: Use the table of values to write the exponential function f(x)=□(v)^x

Solution Steps

To write the exponential function \( f(x) = a \cdot b^x \) using a table of values, we need to determine the base \( b \) and the initial value \( a \). Typically, the initial value \( a \) is the function value when \( x = 0 \). The base \( b \) can be found by examining the ratio of consecutive function values.

1. Identify the initial value \( a \) from the table, which is the function value at \( x = 0 \).
2. Calculate the base \( b \) by dividing the function value at \( x = 1 \) by the function value at \( x = 0 \).
3. Use these values to construct the exponential function.

The initial value \( a \) is the function value when \( x = 0 \). From the table of values, we have: \[ a = f(0) = 2 \]

The base \( b \) is determined by the ratio of the function value at \( x = 1 \) to the function value at \( x = 0 \): \[ b = \frac{f(1)}{f(0)} = \frac{6}{2} = 3.0 \]

Using the values of \( a \) and \( b \), the exponential function can be written as: \[ f(x) = 2 \cdot 3^x \]

Final Answer