The exponential function is a mathematical function that may be useful in determining the growth or decline of a population, money, or price that is expanding or decaying exponentially. Other phenomena that can be analyzed using this function include: Jonathan kept up with the newest studies on bacterial expansion by perusing a recent news item.
He was surprised to learn that just a single bacteria was used in the study. After the first hour, the number of bacteria had increased by one, bringing the total to two. After the first two hours, there were four of them left. The number of microorganisms was steadily growing with each passing hour.
He was pondering the total number of germs over one hundred hours if this trend persisted. When he spoke with his instructor about the matter, the response that he received was on the idea of an exponential function.
Exponential Function
Exponents are involved in the exponential function, which is to be expected given the name of the process. However, notice that an exponential function always has a constant value serving as its base and a variable serving as its exponent.
This cannot be done in reverse (if a function has a variable as the base and a constant as the exponent, then it is a power function but not an exponential function). The following is a list of possible forms that an exponential function may take.
Formal Debilitation Of The Exponential Function
An exponential function is a famous mathematical function that is written in the form of f (x) = axe, where “x” is variable and “a” is a constant that is referred to as the base of the function, & it must be larger than 0.
The Mathematical Form Of The Exponential Form
According to its definition, the most fundamental version of an exponential function is f(x) = bx, where ‘b’ is simply constant and ‘x’ is a variable. One of the most common & essential exponential functions is written as f(x) = ex, where e is referred to as “Euler’s number,” and e equals 2.718…
Looking at the possibilities of various exponential functions from a broader perspective, we can see that an exponential function may incorporate a constant as a multiple of the variable in its power.
In addition, it is also likely to have the form f(x) = p ekx, where p is a constant. Consequently, the following is a list of possible forms for an exponential function:
- f(x) = bx
- f(x) = ax
- f(x) = ABCs
- f(x) = ex
- f(x) = ekx
- f(x) = p ekx
Here, all letters other than ‘x’ are considered constants, and ‘x’ is deemed b. It is necessary to remember that any exponential function’s base must always be a positive integer. Therefore, both b and e are more significant than zero in the tasks described above. Also, b shouldn’t be equal to 1 because if it is, the function f(x) = bx turns into f(x) = 1, which makes the function linear rather than exponential. If b is equal to 1, the process is linear.
The exponential function is generated if the value of a quantity grows at a rate that is exponentially greater than its previous value or falls at a rate that is exponentially more than its last value.
Difference Between Exponential Growth And Exponential Decay
At the start of an exponential growth period, a quantity rises gradually, but subsequently, it begins to expand very quickly. At the beginning of exponential decay, a portion drops at a rapid rate, but after that, it decreases at a much slower pace.
Modeling population expansion, compound interest, determining the time it takes to double an amount, etc., are all applications for the exponential growth formulae. The exponential decay helps model the decline of a population, determining the half-life and other similar tasks.
The function is said to grow at an exponential rate, resulting in a rising graph. The process that represents exponential growth has a chart that slopes downward.
Parameters Of The Exponential Function
The action of an exponential function, denoted by f(x), may be described using a function machine metaphor, much like any other process. This metaphor takes the inputs x and converts them into the outputs f. (x).
The metaphor of a function machine is helpful when it comes to incorporating parameters into a function. The exponential functions f(x) and g(x), discussed before, are two distinct functions; nevertheless, the only thing that differentiates them is the change in the base of the exponentiation, which goes from 2 to 1/2. We could accomplish both duties with a single machine with dials representing the factors that influence how the device operates.