We need to be very careful with the evaluation of exponential functions. The term ‘exponent’ implies the ‘power’ of a number. Clearly then, the exponential functions are those where the variable occurs as a power.An exponential function is defined as- { f(x) = … An Exponential Function is a function of the form y = ab x, where both a and b are greater than 0 and b is not equal to 1.. More About Exponential Function. An exponential function can easily describe decay or growth. In fact, $$g(x)=x^3$$ is a power function. One person takes his interest money and puts it in a box. An exponential model can be found when the growth rate and initial value are known. The general form of an exponential function is y = ab x.Therefore, when y = 0.5 x, a = 1 and b = 0.5. This example is more about the evaluation process for exponential functions than the graphing process. This distinction will be important when inspecting the graphs of the exponential functions. g(x) = … The figure above is an example of exponential decay. A function is evaluated by solving at a specific value. Exponential functions are solutions to the simplest types of dynamic systems, let’s take for example, an exponential function arises in various simple models of bacteria growth. Exponential Decay Exponential decay occurs when a quantity decreases by the same proportion r in each time period t. Here's what that looks like. The number e is important to every exponential function. Definition Of Exponential Function. Exponential Functions. Mathematically, exponential models have the form y = A(r) x, where A is the initial value, and r is the rate of increase (or decrease). Exponential function definition is - a mathematical function in which an independent variable appears in one of the exponents —called also exponential. An exponential function is a mathematical function of the following form: f ( x) = a x. where x is a variable, and a is a constant called the base of the function. Some examples of exponential functions are: Notice that the base of the exponential function, a > 0 , may be greater than or less than one. Thus, $$g(x)=x^3$$ does not represent an exponential function because the base is an independent variable. Example 3 Sketch the graph of $$g\left( x \right) = 5{{\bf{e}}^{1 - x}} - 4$$. The most commonly encountered exponential-function base is the transcendental number e, which is equal to approximately 2.71828.Thus, the above expression becomes: Here's what that looks like. Even though the base can be any number bigger than zero, for example, 10 or 1/2, often it is a special number called e.The number e cannot be written exactly, but it is almost equal to 2.71828.. The function given below is an example of exponential decay. For example, y = 2 x would be an exponential function. 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