Final answer:
The translation from f(x) to g(x) involves a horizontal shift of the graph to the right by 1 unit, as g(x) is f(x) shifted to f(x-1). This changes the constant value in f(x) to a variable one in g(x), thus altering the function's output based on the input x.
Step-by-step explanation:
When comparing the functions f(x)=2 \cdot 3^4 and g(x)=2 \cdot 3^{(x-1)}, the main effect of the translation from f(x) to g(x) is that the constant expression 3^4 in f(x) is replaced by the variable expression 3^{(x-1)} in g(x). This means that whereas f(x) gives the same value for all x, g(x) changes value as x changes. The translation is a horizontal shift of the original function graph in the positive x-direction by 1 unit, as translating a function f(x) to f(x-d) moves the graph d units to the right.
Understanding how exponents work is pivotal in analyzing the behavior of functions. For instance, in the expression (27x^3)(4x^2), raising each factor inside the parentheses by the given power affects everything inside, showing the multiplication rule of exponents that states when raising a power to a power, you multiply the exponents. Similarly, understanding that a negative exponent indicates division (e.g., x^-n = 1/x^n) helps in grasping how functions are inverted or manipulated.