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Exponential growth functions are a class of functions in which the rate of change is directly proportional to the current value of the function. The general formula for such a function is given by \(f(x) = c \cdot a^{kx}\), where c and k are constants and a is called the base of the function. The base, a, is usually a positive number other than 1, since if a = 1, the function would become a constant function rather than an exponential growth function.

The base, a, determines the rapidity of the growth or decay in the function. If the base is between 0 and 1, the function represents exponential decay; if the base is greater than 1, the function represents exponential growth. When working with exponential growth functions, one could choose any base greater than 1, and the function will still exhibit the characteristic exponential growth. The choice of base does not affect the essence of the exponential growth but does affect the rate at which the function grows.

According to the problem statement, we need to show that every exponential growth function can be represented as \(f(x) = c \cdot 3^{kx}\) for appropriate choices of c and k. Let's consider an arbitrary exponential growth function given by \(g(x) = c' \cdot a^{k'x}\), where a is some base greater than 1, and c' and k' are constants. Since \(a > 1\), we can find another constant, say k, such that \(a = 3^k\). If we substitute this expression for a in the general form, we get: \(g(x) = c' \cdot (3^k)^{k'x} = c' \cdot 3^{kk'x}\). Now, if we choose the constant \(c = c' \cdot 3^{ka_1}\) for some constant \(a_1\) where k is defined as \(k = k' + ka_1\), then the function \(g(x)\) can be rewritten as: \(g(x) = c \cdot 3^{kx}\). Since we started with an arbitrary exponential growth function and were able to derive the desired form \(f(x) = c \cdot 3^{kx}\), it can be concluded that every function with exponential growth can be represented by a formula of this form, with the appropriate choice of constants c and k.