Exponential Functions

An exponential function is a Mathematical function in form f (x) = a, where “x” is a variable and “a” is a constant which is called the base of the function and it should be greater than 0. The most commonly used exponential function base is the transcendental number e, which is approximately equal to 2.71828. In this article, let us discuss in detail about the exponential function formulas, rules, properties, graphs, and examples.

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Exponential Function Formula

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

An exponential curve can 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 exponential decay.

Exponential Growth

In Exponential Growth, the quantity increases very slowly at first, and then rapidly. The rate of change increases over time. The rate of growth becomes faster as time passes. The rapid growth meant to be an “exponential increase”. The formula to define the exponential growth is:

y = a ( 1+ r )x

Where r is the growth percentage.

Exponential Decay

In Exponential Decay, the quantity decreases very rapidly at first, and then slowly. The rate of change decreases over time. The rate of change becomes slower as time passes. The rapid growth meant to be an “exponential decrease”. The formula to define the exponential growth is:

y = a ( 1- r )x

Where r is the decay percentage.

Exponential Functions Examples

The examples of exponential functions are:

  • f(x) = 2x
  • f(x) = 1/ 2x = 2-x
  • f(x) = 2x+3
  • f(x) = 0.5x

Exponential Function Properties

The exponential graph of a function represents the exponential function properties.

Let us consider the exponential function, y=2x

The graph of function y=2x is shown below. First, the property of the exponential function graph when the base is greater than 1.

Exponential Functions

Exponential Function Graph for y=2x

The graph passes through the point (0,1).

  • The domain is all real numbers.
  • The range is y>0.
  • The graph is increasing.
  • The graph is asymptotic to the x-axis as x approaches negative infinity
  • The graph increases without bound as x approaches positive infinity
  • The graph is continuous
  • The graph is smooth

Exponential Functions

Exponential Function Graph y=2-x 

The graph of function y=2-x is shown above. The properties of the exponential function and its graph when the base is between 0 and 1 are given.

  • The line passes through the point (0,1).
  • The domain includes all real numbers.
  • The range is of y>0.
  • It forms a decreasing graph
  • The line in graph above is asymptotic to the x-axis as x approaches positive infinity.
  • The line increases without bound as x approaches negative infinity.
  • It is a continuous graph.
  • It forms a smooth graph.

Exponential Function Rules

Some important exponential rules are given below:

If a>0, and  b>0, the following hold true for all the real numbers x and y:

    • ax ay = ax+y
    • ax/ay = ax-y
    • (ax)y = axy
    • axbx=(ab)x
    • (a/b)x= ax/bx
    • a0=1
    • a-x= 1/ ax

Exponential Series

Exponential Functions

For Complex exponential function and exponential function formula, visit www.byjus.com for a detailed explanation.

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