Final answer:
The function f(a+b) where f(x) = 2^x for any positive numbers a and b is equivalent to 2^(a+b), which simplifies to 2^a * 2^b using the properties of exponents.
Step-by-step explanation:
If we have a function f(x) = 2^x, and a and b are any positive numbers, then f(a+b) would be equivalent to applying the function to the sum of a and b, resulting in 2^(a+b). In exponents, when we raise a number to the power of two expressions added together, this is the same as taking the product of the number raised to each expression individually. Therefore, if we use the properties of exponents, we can say:
f(a+b) = 2^(a+b) = 2^a * 2^b
This is because the logarithm of a product of two numbers is the sum of the logarithms of the two numbers. In other words, log xy = log x + log y and similarly for natural logarithms ln xy = ln x + ln y. So by using the rules of exponents, f(a+b) simplifies to the product of f(a) and f(b), or 2^a * 2^b.