Question

answer the questions below to determine what kind of function is depicted in the table below.

x	f(x)
1	63
2	189
3	567
4	1701

this function is
because

Explanation:

Step1: Check if it's linear (constant difference)

Calculate the differences between consecutive \( f(x) \) values:
\( 63 - 21 = 42 \), \( 189 - 63 = 126 \), \( 567 - 189 = 378 \), \( 1701 - 567 = 1134 \).
Differences are not constant, so not linear.

Step2: Check if it's exponential (constant ratio)

Calculate the ratios of consecutive \( f(x) \) values:
\( \frac{63}{21} = 3 \), \( \frac{189}{63} = 3 \), \( \frac{567}{189} = 3 \), \( \frac{1701}{567} = 3 \).
Ratios are constant (\( r = 3 \)), and when \( x = 0 \), \( f(0) = 21 \) (the initial value \( a = 21 \)). So the function follows \( f(x) = 21 \cdot 3^x \), which is exponential.

Answer:

This function is an exponential function because the ratio of consecutive \( f(x) \) values is constant (equal to 3), indicating exponential growth with an initial value of 21 and a common ratio of 3.