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 x becomes an exponent. Mathematically, exponential equations are written as:

y = a ^ x where a > 0 and a \neq 1

Here, a is the base of an exponential function, y is a dependent variable and x 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 a is not equal to 1 or 0 because no matter what the value of x is, they remain constant. It can be expressed as:

1 ^ x = 1      0 ^ x = 0 {where x 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 f(x) = 2 ^ x. The table of values for this exponential function is given below:

xy
-31/8
-21/4
-11/2
01
12
24
38
 The graph of this exponential function is given below:
Parent Function graph

Properties of Exponential Functions

You can see the above graph represents an exponential function with a base greater than 1. The parent exponential function y = 2 ^ x has the following properties:
  • The point (0, 1) is the part of the graph which means that the graph passes through the point (0, 1)
  • 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 y is equal to all positive real numbers.
  • The graph increases and is asymptotic to the negative side of the  x-axis as x approaches - \infty
  • The graph increase without limit as x approaches +\infty
  • 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 y = \frac{1}{2} ^ x. The table of values of this exponential function is given below:

xy
-38
-24
-12
01
11/2
21/4
31/8

The graph of this exponential function is given below:

Exponential function (base less than 1)

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 (0 , 1) 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. y > 0. Again, this property is the same for both types of exponential functions
  • The graph decreases and is asymptotic to the positive side of x - axis as x reaches +\infty
  • The graph increases without limit as x reaches -\infty
  • The graph of this exponential function is also smooth and continuous
  • The graph is asymptotic to the x-axis as x approaches to infinity

 

Examples

Example 1

Now, we will see how we can graph exponential functions other than the parent exponent function. Consider the following example having a base greater than 1.

y = 3 ^ {x +1}

To graph this function, first we need to construct a table of values for this function. Remember that this is not a simple exponential function. You need to be careful in calculating the y values because the operation x + 1 is involved.

xy
-21/3
-11
03
19
227

Now, that we have obtained  the table of values for the function, we can easily construct the graph by plotting the values in x - y plane.

Example 1 - Exponential functions

Example 2

Now, we will see an example of an exponential function having base less than 1. Consider the following exponential function:

(\frac{1}{2}) ^ {x+1}

Before graphing this exponential function, we need to construct a table of values first like this:

xy
-22
-11
01/2
11/4
21/8

Using the above table of values, we will construct the graph of an exponential function like this:

Example 2 - Exponential functions

 

Exponential Functions Word Problems

We already have discussed in the article that the exponential functions are applicable to our daily lives. Let us see how to solve exponential growth and decay word problems.

Example 1

Sam bought a car of worth 15000 dollars. The car depreciates at an annual rate of 7%. What will be the price of car after six years?

Solution

The initial price of the car is 15000 dollars. The car depreciates at an annual rate of 7%, so (1 - r) will be equal to (1 - 0.07). Using this information, the exponential function will be written as:

f(x) = 15000 (1 - r ) ^ t

  f(x) = 15000(1 - 0.07) ^ 6

= 15000(0.93) ^ 6

= 15000 (0.6469)

= 9703.5

Hence, the price of the car after 6 years will be 9703.5 dollars.

 

Example 2

According to a census, the population of the city in 2014 was 150,000. The population is expected to increase by 11% annually. What will be the population of the city in 2019.

Solution

According to the above problem, we have got the following information:

Initial population of the city = a = 150,000

Annual increase in the population = r = 11% = 0.11

Number of years = t = 5

The function which will be used to compute the growth in the population is given below:

a (1 + r) ^ t

Solve this equation by plugging in the values from the problem:

=150,000 ( 1 + 0.11 ) ^ 5

=150,000 (1.11) ^ 5

=150,000 (1.685)

= 252,750

Hence, the population of the city after 5 years will be 252, 750.

 

Example 3

The half life of a radioactive element is 3.2 hours. If a scientist has 92 grams of the element now, how many grams will be left after 12.8 hours?

Solution

This is a half life problem and will solved differently than exponential growth and decline problems. We have obtained the following information from the above problem:

The half life of the radioactive element = 3.2 hours

Number of half lives = \frac {12.8}{3.2} = 4

Take the exponent of \frac {1}{2} to compute the value which will be later by the initial amount to get the amount left after 12.8 hours.

= \frac {1}{2} ^ 4 = 0.0625

Number of grams left after 4 half lives = 92 \cdot 0.0625

= 5.75 grams

Hence, after 12.8 hours the scientist will have 5.75 grams of radioactive element.

 

Example 4

The half life of a radioactive element is 13 hours. If Sarah had 148 grams of the element now, how many grams will be left after 72 hours?

Solution

From the above problem, we have obtained the following information:

Half life of the radioactive element = 13 hours

Number of half lives will be calculated by dividing total hours passed by the half life of the radioactive element.

Number of half lives = \frac {72}{13} = 5.53

We need to multiply \frac{1}{2} 5.53 times first using a calculator and then multiply the amount obtained with the initial amount of radioactive element to calculate the number of grams of radioactive element left in 5.53 half lives. To simplify, we have divided this final step into two parts.

= \frac {1}{2} ^ {5.53} = 0.0216

Number of grams left in 5.53 half lives = 148 .  0.0216

= 3.196 grams

Hence, after 72 hours 2.196 grams of the radioactive element will be left.

 

Example 5

The half life of a radioactive element is 7 hours. If George has 90 milli grams of radioactive element now, how many milligrams will be left after 33.6 hours

Solution

The information obtained from the above problem is below:

The original amount of the radioactive element = 90 milligrams

Number of hours passed = 33.6 hours

The half life of radioactive element = 7 hours

Number of half lives will be calculated by the time passed, i.e. 33.6 hours with the half life of the radioactive element which is 7 hours.

Number of half lives = 4.8

To calculate the amount left after 33.6 hours, first, we will multiply \frac{1}{2} 4.8 times.

= \frac {1}{2} ^ {4.8} = 0.0358

Now, we can easily calculate the amount left after 4.8 half lives have passed by multiplying the amount left after 33.6 hours by the initial amount George had.

Number of milligrams left after 4.8 half lives = 0.0358 . 90

= 3.22 milligrams

Hence, after 33.6 hours 3.22 milligrams of the radioactive element will be left.

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

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