Question

example 3
what is the written form of the function represented by the table?

x	f(x)
1	12
2	36
3	108
4	324

Explanation:

Step1: Check if it's exponential

An exponential function has the form \( f(x) = a \cdot b^x \). Let's check the ratios between consecutive \( f(x) \) values.
\( \frac{12}{4} = 3 \), \( \frac{36}{12} = 3 \), \( \frac{108}{36} = 3 \), \( \frac{324}{108} = 3 \). So the common ratio \( b = 3 \).

Step2: Find the initial value \( a \)

When \( x = 0 \), \( f(0) = a \cdot b^0 = a \cdot 1 = a \). From the table, when \( x = 0 \), \( f(0) = 4 \), so \( a = 4 \).

Step3: Write the function

Using the exponential function form \( f(x) = a \cdot b^x \) with \( a = 4 \) and \( b = 3 \), we get \( f(x) = 4 \cdot 3^x \).

Answer:

The function is \( f(x) = 4 \cdot 3^x \)