Question

find the equation of the exponential function.

x	f(x)
1	5·3
2	5·(3)²
3	5·(3)³

show your work here

hint: to add an exponent (xⁿ), type \exponent\ or press \^\.

f(x) =

Explanation:

Step1: Recall exponential function form

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

Step2: Determine \( a \) from \( x = 0 \)

When \( x = 0 \), \( f(0) = 5 \). Substituting into the general form: \( f(0) = a \cdot b^0 \). Since \( b^0 = 1 \), we get \( a = 5 \).

Step3: Determine \( b \) from the table

Looking at the table, when \( x = 1 \), \( f(1) = 5 \cdot 3 \); when \( x = 2 \), \( f(2) = 5 \cdot 3^2 \); when \( x = 3 \), \( f(3) = 5 \cdot 3^3 \). The base \( b \) is 3 (since the exponent is \( x \) and the coefficient of 5 is \( 3^x \)).

Step4: Write the exponential function

Substituting \( a = 5 \) and \( b = 3 \) into the general form, we get \( f(x) = 5 \cdot 3^x \).

Answer:

\( f(x) = 5 \cdot 3^x \)