Final answer:
A recursive rule for the function g(x) = 5(2)^x is given by g(x) = 2 * g(x-1) for x > 0, with the base case being g(0) = 5. This rule utilizes the property that multiplying powers with the same base is the same as adding exponents.
Step-by-step explanation:
Recursive Rule for the Function g(x)
Let's define a recursive rule for the function g(x) = 5(2)^x. A recursive rule builds each term based on the previous term. For this exponential function, let’s consider that g(0) = 5(2)^0, which is the base case, and equals 5, since any number to the power of 0 is 1. Now for x > 0,
g(x) = 2 * g(x-1),
meaning each term g(x) is twice the previous term g(x-1). This is because raising 2 to the power of x and then to the power of x-1 effectively just subtracts one from the exponent, following the property that (a^m)^n = a^(m*n). For example, 5^3 * 2 = 5^3 * 5^1 which is 5^(3+1) or 5^4, showing that multiplying by an exponential base is akin to increasing the exponent by one.