In this article, we will learn exponential functions in detail. We will see how to construct a table of values of these mathematical functions and graph them using that table. We will also study properties of the exponential functions and solve word problems related to exponential growth and exponential decay. So, let us get started.

After linear functions, exponential functions are the most popular functions in mathematics which are also used in our daily life. For example, we use exponential functions to compute population growth or decline. In exponential functions, the independent variable becomes an exponent. Mathematically, exponential equations are written as:

where and

Here, a is the base of an exponential function, is a dependent variable and is an independent variable. The base of an exponential function should be a positive number because a if the base is a negative number, it will unnecessarily complicate the function. Exponential functions are the inverses of logarithms. The fixed term is not equal to 1 or 0 because no matter what the value of is, they remain constant. It can be expressed as:

{where can be any value}

If you want to calculate the population growth, exponential function increases quickly. Similarly, if you are computing a population decline, exponential functions decrease swiftly.

Let us make the graph of the exponential functions. The parent exponential function is . The table of values for this exponential function is given below:

x | y |
---|---|

-3 | 1/8 |

-2 | 1/4 |

-1 | 1/2 |

0 | 1 |

1 | 2 |

2 | 4 |

3 | 8 |

## Properties of Exponential Functions

- The point is the part of the graph which means that the graph passes through the point
- The domain of the exponential function having the base greater than 1 is equal to set of all real numbers
- The range of the exponential function having a base greater than 1 is set of all real numbers greater than 0. It means is equal to all positive real numbers.
- The graph increases and is asymptotic to the negative side of the -axis as approaches
- The graph increase without limit as approaches
- The graph of such exponential function is smooth and continuous

So far, we have discussed the graph and properties of the exponential function having the base greater than 1. Now, we will see what the properties of the exponential functions are if their base is less than 1. Let us consider an exponential function . The table of values of this exponential function is given below:

x | y |
---|---|

-3 | 8 |

-2 | 4 |

-1 | 2 |

0 | 1 |

1 | 1/2 |

2 | 1/4 |

3 | 1/8 |

The graph of this exponential function is given below:

Now, you can tell the difference between the exponential functions with base greater than and less than 1. The properties of exponential functions having base less than 1 are given below:

Properties of Exponential Function when the base is less than 1

- Point is part of the graph which means that the graph passes through the point (0,1)
- The domain of this exponential function is also equal to set of all real numbers
- The range of this exponential function is all positive real numbers, i.e. . Again, this property is the same for both types of exponential functions
- The graph decreases and is asymptotic to the positive side of - axis as reaches
- The graph increases without limit as reaches
- The graph of this exponential function is also smooth and continuous
- The graph is asymptotic to the x-axis as x approaches to infinity

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