Final answer:
To simplify the function f(x) = 1/3(81)^3x/4, we identify 81 as 3^4 and then apply exponent rules which lead to the simplified form f(x)=3^(3x-1). The base indicates exponential growth, and the exponent determines the growth rate and vertical scaling. Additional analysis would involve graphing to observe its behavior.
Step-by-step explanation:
To simplify the function f(x)=1/3(81)^3x/4, we must first recognize that 81 is a power of 3, specifically 3^4. Thus, when raised to the 3x/4 power, we multiply these exponents to get 3^(4*3x/4) which simplifies to 3^(3x). Our function now looks like f(x)=(1/3)*3^(3x). By properties of exponents, this can further simplify to f(x)=3^(3x-1).
Key aspects of this function include the base of the exponential (which is 3), indicating that the growth is exponential, and the exponents indicate how quickly this growth occurs. The term '-1' in the exponent shows a vertical scaling factor of 1/3 on the base function 3^x.
To analyze the function in more detail, one could use a graphing calculator or computer software to visualize the growth pattern. However, it should be noted that in this case, a graphing calculator can't solve for x using the zero function as we don't have an equation set to zero.