Exponential Functions

• An exponential function in Mathematics can be defined as a Mathematical function is in form f(x) = ax, where “x” is the variable and where “a” is known as a constant which is also known as the base of the function and it should always be greater than the value zero. 

• The most commonly used exponential function base is the transcendental number denoted by e, which is approximately equal to the value of 2.71828.

• An exponential function formula can be defined by f(x) = ax, where the input variable is denoted as x occurs as an exponent. The exponential curve depends on the exponential function and it depends on the value of the x.

Exponential Function Formula

The exponential function is an important mathematical function, the exponential function formula can be written in the form of:

Where the value of a > 0 and the value of a is not equal to 1.

X can be any real number.

If the value of the variable is negative, the function is undefined for (range of x) -1 < x < 1.

Here, “x” can be defined as a variable and “a” is a constant, which is defined as the base of the function.

But, mostly the base of the exponential function is encountered by the transcendental number “e”, which is approximately equal to the value of 2.71828.

An exponential curve can basically grow or decay depends on the exponential function. Any quantity that grows or decays by a fixed per cent at regular intervals should possess either exponential growth or should possess exponential decay.

Exponential Growth

The quantityIn Exponential Growth increases very slowly at first, and then it increases rapidly. The rate of change increases over time. As time passes,the rate of growth becomes faster. The rapid growth can also be known as an “exponential increase”. The formula to define the exponential growth can be defined by the formula:

y = a (1+ r )x

where r is equal to the growth percentage.

Exponential Decay

The quantity in Exponential decay decreases very rapidly at first, and then it decreases slowly. The rate of change generally decreases over time. The rate of change becomes slower as the time passes. The rapid growth can also be known as an “exponential decrease”. The formula to define the exponential growth is denoted by :

y = a (1- r )x

where the variable r is equal to the decay percentage.

Overview of Exponential Function

To form an exponential function, we generally let the independent variable be the e( known as the exponent). A simple example is the function using exponential function graph

Function f(x)=2x

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As we can see in the given exponential function graph of f(x) that the exponential function increases rapidly. 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. 

An exponential function can easily describe decay or growth. The function given below is an example of exponential decay.

g(x) =\[\mathrm{\dfrac{1}{2}}^{x}\]

It gets rapidly smaller as the value of x increases, as illustrated by its graph that is given below.

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In the given exponential function graph example above the exponential growth of the function f(x),the function doubles every time when you add one to its input x. In the exponential decay of the function g(x), the function shrinks in half every time you add one to its input variable x. 

Parameters of the Exponential Function

As with any function, the action of a given exponential function f(x) can be captured by the function machine metaphor, that takes inputs  for variable x and transforms them into the outputs f(x).

The function machine metaphor is useful for introducing the parameters into a function. The above exponential functions f(x) and g(x) are two different functions, but these functions differ only by the change in the base of exponentiation that is from 2 to 1/2. 

Properties of Exponential Functions

Some of the important properties of exponential functions are given below:

If the value of a>0, and the value of b>0, then the following hold true for all the real numbers m and n:

• \[a^m\cdot a^n=a^{m+n}\]

• \[\dfrac{a^m}{a^n}=a^{m-n}\]

• (am)n=amn

• \[a^m b^m={(ab)}^{m}\]

• \[\mathrm{(\dfrac{a}{b})}^m=\dfrac{a^m}{b^m}\]

• \[a^0=1\]

• \[a^{-m}=\dfrac{1}{a^m}\]

Examples of Exponential Functions-

The exponential function  examples are listed below:

• f(x) = 3x

• f(x) =\[\dfrac{1}{3^x}=3^{-x}\]

• f(x) = 5x+3

• f(x) = 0.6x

Questions to be solved:

Question 1)Simplify the given exponential equation 3x - 3x+1

Solution)We have the given exponential equation: 3x - 3x+1

By using the properties of exponential function: am an = am+n

Hence, 3x+1 can be written as 3x. 3

 Thus the given equation can also be written as:

3x-3x+1 =3x-3x. 3

Now, factor out the term 3xfrom the equation,

3x-3x+1 =3x-3x. 3 = 3x(1-3)

3x-3x+1 = 3x(-2)

3x-3x+1 = – 2(3)x

For x=0,

We get,

30-30+1=-2(3)0

1-3=-2(1)
-2=-2

Therefore, the simplification of the given exponential equation  3x-3x+1 is equal to  – 2(3)x which is the answer.