What is a Functional Exponent?

What is a Functional Exponent?

Definition

A Functional Exponent is a function in the form of f(x) = ax, where x is a variable and a is a constant, and it is known as the base of the function and should be greater than 0. Functional exponent is also called exponential function.

Forms of Functional Exponent

  • f(x) = ax
  • f(x) = bax , where b is also a constant
  • f(x) = ex
  • f(x) = peqx , where p is also a constant

Examples of Functional Exponent

The most common example of the functional exponent is ex, where the value of e is 2.718. Some other examples are as follows:-

  • 5e15x
  • (1/2)x
  • 7(8)-0.7x, etc.

Properties of Functional Exponent

  • a0 = 1
  • (am)n = am x n
  • am x an = a(m + n)
  • am / an = a(m – n)
  • (a x b)m = am x bm
  • (a / b)m = am / bm
  • a-m = 1/a

Relation between Functional Exponent and Logarithmic Functions

If there is any functional exponent that is in the form of:-

px = b

Then, it can be written as:-

log p b = x

So, the relation between Functional Exponent and Logarithmic Functions is:-

px = b => log p b = x

Some Graphs of Functional Exponent

1. Graph of 5x

Let, f(x) = 5x . So, some values of f(x) will be-

values of 5x in Functional Exponent

So, the Graph of 5x will be:-

graph of 5x in Functional Exponent

2. Graph for (1/2)x

Let f(x) = (1/2)x. So, some values of f(x) will be-

values of (1/2)x in exponential functions

So, the Graph of (1/2)x will be:-

graph of (1/2)x in exponential functions

3. Graph of ex

Let f(x) = ex. So, some values of f(x) will be-

values of ex in functional exponents

Value of e = 2.718

So, the graphs of the ex will be-

graph of ex in functional exponents

Derivative of Functional Exponent or Exponential Function

  • d/dx (ex) = ex
  • d/dx (ax) = ax log a

Integration of Functional Exponents

  • ∫ ex dx = ex + C
  • ∫ ax dx = (ax / log a) + C

Where C is a constant.

Series Expansion of ex

series of ex in exponential functions

So, What is a functional exponent? The answer is functional exponent is the same as the exponential functions that we see in Mathematics, it’s just that we are calling exponential functions a functional exponent. We have seen most of the required things anyone needs to know about functional exponents. We hope this article on the Exponetinial Function Helps you.

If you have any questions/doubts in mind, please use the comments below.

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