Lesson Explainer: Exponential Functions Mathematics

In this explainer, we will learn how to identify, write, evaluate, and analyze exponential functions.

Definition: Exponential Function

An exponential function is a function with a rule of the form ๐‘“(๐‘ฅ)=๐‘,๏— where the constant ๐‘ is the base and the variable ๐‘ฅ is the exponent. The constant ๐‘ is a real number such that ๐‘>0 and ๐‘โ‰ 1 , and the variable ๐‘ฅ can be any real number.

To analyze an exponential function with the rule ๐‘“(๐‘ฅ)=๐‘๏—, letโ€™s begin by finding ๐‘“(๐‘ฅ) for some positive integer values of ๐‘ฅ.

๐‘ฅ1234
๐‘“(๐‘ฅ)๐‘=๐‘๏Šง๐‘=๐‘โ‹…๐‘๏Šจ๐‘=๐‘โ‹…๐‘โ‹…๐‘๏Šฉ๐‘=๐‘โ‹…๐‘โ‹…๐‘โ‹…๐‘๏Šช

We can see that the previous value of ๐‘“(๐‘ฅ) is multiplied by ๐‘ every time ๐‘ฅ increases by 1. That is,

  • the value of ๐‘“(2) is ๐‘“(1)โ‹…๐‘,
  • the value of ๐‘“(3) is ๐‘“(2)โ‹…๐‘,
  • the value of ๐‘“(4) is ๐‘“(3)โ‹…๐‘.

Property: Relationship between ๐‘“(๐‘ฅ) and ๐‘“(๐‘ฅ โˆ’ 1) for the Exponential Function ๐‘“(๐‘ฅ) = ๐‘๐‘ฅ

For the exponential function ๐‘“(๐‘ฅ)=๐‘๏—, the value of ๐‘“(๐‘ฅ) is always the product of ๐‘“(๐‘ฅโˆ’1) and ๐‘, which means that ๐‘ is always the quotient of ๐‘“(๐‘ฅ) and ๐‘“(๐‘ฅโˆ’1). That is, ๐‘“(๐‘ฅ)=๐‘“(๐‘ฅโˆ’1)โ‹…๐‘๐‘=๐‘“(๐‘ฅ)๐‘“(๐‘ฅโˆ’1).

The above property is true not only for positive integer values of ๐‘ฅ, but for any real number value of ๐‘ฅ. Thus, we can often use the relationship between ๐‘, ๐‘“(๐‘ฅ), and ๐‘“(๐‘ฅโˆ’1) to find the value of ๐‘ from a graph or a table.

Consider the following graph, which shows ๐‘ฆ=๐‘“(๐‘ฅ)=๐‘๏— for a specific value of ๐‘. Notice that the graph intersects the ๐‘ฆ-axis at 1. This will be the case for any real number value of ๐‘ such that ๐‘>0 and ๐‘โ‰ 1.

We can find the value of ๐‘ in the function that the graph represents by selecting two points that the graph passes through with ๐‘ฅ-coordinates that differ by 1. It does not matter which points we choose, but both should have ๐‘ฅ- and ๐‘ฆ-coordinates that are easy to identify. In this case, it is easiest for us to select points with ๐‘ฅ-coordinates that are consecutive integers.

It is clear that the graph passes through the points (0,1) and (โ€“1,2), so we will choose these two points to find the value of ๐‘. Based on the coordinates of the points, we know that ๐‘“(0)=1๐‘“(0โˆ’1)=๐‘“(โˆ’1)=2.and

Remember that ๐‘ is always equal to the quotient of ๐‘“(๐‘ฅ) and ๐‘“(๐‘ฅโˆ’1). Therefore, ๐‘=๐‘“(๐‘ฅ)๐‘“(๐‘ฅโˆ’1)=๐‘“(0)๐‘“(โˆ’1)=12

So, the graph represents the function ๐‘“(๐‘ฅ)=๏€ผ12๏ˆ๏—.

Next, letโ€™s look at a table representing the function ๐‘”(๐‘ฅ)=๐‘๏— for a specific value of ๐‘.

๐‘ฅ1234
๐‘”(๐‘ฅ)24816

Since the values of ๐‘ฅ in the table differ by 1, we could use the same procedure as above to determine the value of ๐‘, finding either the quotient of 4 and 2, the quotient of 8 and 4, or the quotient of 16 and 8. However, this time, we will instead substitute the first pair of ๐‘ฅ- and ๐‘ฆ-values in the table into the rule ๐‘”(๐‘ฅ)=๐‘๏—. Substituting 1 for ๐‘ฅ and 2 for ๐‘”(๐‘ฅ) gives us 2=๐‘,2=๐‘,๏Šงor which tells us that the value of ๐‘ must be 2. Thus, the table represents the function ๐‘”(๐‘ฅ)=2๏—.

So far, we have looked at only exponential functions in the form ๐‘“(๐‘ฅ)=๐‘๏—, but an exponential function can be transformed in the usual manner. While the value of the base must always be a positive real number that is not equal to 1, the exponent can consist of any number of linear terms, as long as at least one of them contains a variable. Also, sometimes the exponential term is multiplied by a constant, or a constant is added to it. Some examples of transformed exponential functions include:

  • ๐‘”(๐‘ฅ)=4โ‹…2๏—,
  • โ„Ž(๐‘ฅ)=๐‘’+1๏—,
  • ๐‘˜(๐‘ฅ)=9๏Šช๏—๏Šฐ๏Šฉ.

Just as with exponential functions in the form ๐‘“(๐‘ฅ)=๐‘๏—, we can see that in each of these examples, the exponent contains a variable, is raised, and is to the right of the base:

  • for ๐‘”(๐‘ฅ), the base is 2, and the exponent is ๐‘ฅ,
  • for โ„Ž(๐‘ฅ), the base is ๐‘’, and the exponent is ๐‘ฅ,
  • for ๐‘˜(๐‘ฅ), the base is 9, and the exponent is 4๐‘ฅ+3.

We can also find the equations of transformed exponential functions such as these from graphs and tables, which we will do in some of the problems that follow. We will begin, though, with a problem that asks us to identify an exponential functionโ€™s base and exponent.

Example 1: Identifying the Base and the Exponent of an Exponential Function

What are the base and exponent of the function ๐‘“(๐‘ฅ)=5๏—๏Šฑ๏Šซ?

Answer

Remember that for an exponential function with the rule ๐‘“(๐‘ฅ)=๐‘๏—, the base is the constant ๐‘, and the exponent is the variable ๐‘ฅ. We know that ๐‘ is a real number such that ๐‘>0 and ๐‘โ‰ 1 and that ๐‘ฅ can be any real number.

Our exponential function has been transformed. In it, we have the variable binomial expression ๐‘ฅโ€“5 raised and to the right of 5. Therefore, we know that the base is 5 and that the exponent is ๐‘ฅโ€“5.

In the next problem, we are given a table of values for an exponential function and are asked to find the functionโ€™s equation.

Example 2: Writing an Exponential Equation from a Table of Values

Write an exponential equation in the form ๐‘ฆ=๐‘๏— for the numbers in the table.

๐‘ฅ245
๐‘ฆ916812562431024

Answer

Recall that for an exponential equation in the form ๐‘ฆ=๐‘๏—, the base is the constant ๐‘, and the exponent is the variable ๐‘ฅ. We know that ๐‘ is a real number such that ๐‘>0 and ๐‘โ‰ 1 and that ๐‘ฅ can be any real number.

Here, since we are asked to write an exponential equation in the form ๐‘ฆ=๐‘๏— for the numbers in the table, we must find the value of the base, ๐‘, to solve the problem.

Letโ€™s begin by substituting one of the pairs of ๐‘ฅ- and ๐‘ฆ-values from the table into the equation ๐‘ฆ=๐‘๏—.

Substituting 2 for ๐‘ฅ and 916 for ๐‘ฆ gives us 916=๐‘.๏Šจ

Now, to solve for ๐‘, we can take the square root of both sides of the equation. A square root can be positive or negative, but since ๐‘ must be a positive real number, we can ignore the negative root, giving us ๏„ž916=โˆš๐‘.๏Šจ

Remember that to find the square root of a fraction, we take the square root of both the numerator and the denominator, so our equation can be simplified as follows: โˆš9โˆš16=โˆš๐‘34=๐‘.๏Šจ

Thus, since the value of ๐‘ is 34, an exponential equation in the form ๐‘ฆ=๐‘๏— for the numbers in the table is ๐‘ฆ=๏€ผ34๏ˆ๏—.

Check

We can verify our answer by using another method to solve for the base, ๐‘. To use this method, we will substitute two other pairs of ๐‘ฅ- and ๐‘ฆ-values from the table into the equation ๐‘ฆ=๐‘๏—. First, we will substitute 4 for ๐‘ฅ and 81256 for ๐‘ฆ to get 81256=๐‘.๏Šช

Next, we will substitute 5 for ๐‘ฅ and 2431024 for ๐‘ฆ to get 2431024=๐‘.๏Šซ

Now we can divide the second equation by the first equation. Doing so gives us ๏Šจ๏Šช๏Šฉ๏Šง๏Šฆ๏Šจ๏Šช๏Šฎ๏Šง๏Šจ๏Šซ๏Šฌ๏Šซ๏Šช=๐‘๐‘.

This equation can be rewritten as 2431024โ‹…25681=๐‘๏Šซ๏Šฑ๏Šช and simplified to 34=๐‘.

This confirms that the value of ๐‘ is 34 and that an exponential equation in the form ๐‘ฆ=๐‘๏— for the numbers in the table is ๐‘ฆ=๏€ผ34๏ˆ๏—.

Note that in the process of checking our answer, we used the quotient rule, which is one of the properties of exponents.

Definition: Quotient Rule

The quotient rule states that when dividing exponential expressions with the same base, we keep the base and find the difference of the exponents. That is, ๐‘๐‘=๐‘,๏‰๏Š๏‰๏Šฑ๏Š where ๐‘ is the base and ๐‘š and ๐‘› are the exponents.

For example, we would use the quotient rule to find that 55=5=5๏Šฎ๏Šช๏Šฎ๏Šฑ๏Šช๏Šช.

In the problem that follows, we are again given a table of values for an exponential function and are asked to find the functionโ€™s equation. This time, however, the equation is in a form other than ๐‘ฆ=๐‘๏—.

Example 3: Writing an Exponential Equation from a Table of Values

Write an exponential equation in the form ๐‘ฆ=๐‘Ž(๐‘)๏— for the numbers in the table.

๐‘ฅ0123
๐‘ฆ186223

Answer

In this problem, we are asked to write an exponential equation in the form ๐‘ฆ=๐‘Ž(๐‘)๏— for the numbers in the table, so we must find the values of ๐‘Ž and ๐‘ to solve the problem. Letโ€™s begin by substituting one of the pairs of ๐‘ฅ- and ๐‘ฆ-values from the table into the equation ๐‘ฆ=๐‘Ž(๐‘)๏—.

Substituting 0 for ๐‘ฅ and 18 for ๐‘ฆ gives us 18=๐‘Ž๏€น๐‘๏….๏Šฆ

Since any base raised to the power 0 is equal to 1, the equation can then be simplified to 18=๐‘Ž(1),18=๐‘Ž.or

Now that we know that the value of ๐‘Ž is 18, we can find the value of ๐‘. To do so, we will again substitute one of the pairs of ๐‘ฅ- and ๐‘ฆ-values from the table into the equation ๐‘ฆ=๐‘Ž(๐‘)๏—. This time, we will substitute 1 for ๐‘ฅ and 6 for ๐‘ฆ to get 6=๐‘Ž๏€น๐‘๏…,6=๐‘Ž๐‘.๏Šงor

Recall that we just found the value of ๐‘Ž to be 18, so we can also substitute 18 into the equation for ๐‘Ž, which gives us 6=18๐‘.

Next, dividing both sides of the equation by 18, we arrive at 13=๐‘.

Thus, since the value of ๐‘Ž is 18 and the value of ๐‘ is 13, an exponential equation in the form ๐‘ฆ=๐‘Ž(๐‘)๏— for the numbers in the table is ๐‘ฆ=18๏€ผ13๏ˆ๏—.

Check

To verify our answer, we can substitute the last two ๐‘ฅ-values from the table into our equation to make sure they give us the correct ๐‘ฆ-values. First, substituting 2 for ๐‘ฅ and then simplifying gives us ๐‘ฆ=18๏€ผ13๏ˆ=18๏€ผ19๏ˆ=2.๏Šจ

Next, substituting 3 for ๐‘ฅ and then simplifying, we get ๐‘ฆ=18๏€ผ13๏ˆ=18๏€ผ127๏ˆ=23.๏Šฉ

Since both ๐‘ฅ-values give us the correct ๐‘ฆ-value, we can be confident that our equation is correct.

Note that in the process of solving for a, we used one of the properties of exponents, called the zero exponent rule.

Definition: Zero Exponent Rule

The zero exponent rule states that any base raised to the power 0 is equal to 1. That is, ๐‘=1,๏Šฆ where ๐‘ is the base.

For example, we would use the zero exponent rule to find that 3=1๏Šฆ.

Now we will write the equation of an exponential function from its graph.

Example 4: Forming the Equation of an Exponential Function from Its Graph

Observe the given graph, and then answer the following questions

  • Find the ๐‘ฆ-intercept in the shown graph.
  • As this graph represents an exponential function, every ๐‘ฆ-value is multiplied by ๐‘ when ๐‘ฅ increases by ฮ”๐‘ฅ. Find ๐‘ for ฮ”๐‘ฅ=1.
  • Find the equation that describes the graph in the form ๐‘ฆ=๐‘Ž๐‘๏‘๏ซ๏‘.

Answer

Part 1

Letโ€™s begin by finding the graphโ€™s ๐‘ฆ-intercept. We can see that the graph passes through the point (0,10) on the ๐‘ฆ-axis. In other words, it passes through the ๐‘ฆ-axis at a ๐‘ฆ-value of 10.

Hence, 10 is the ๐‘ฆ-intercept of the shown graph.

Part 2

Now letโ€™s find the value of ๐‘ in the equation of the graph. In the problem, we are told that the graph represents an exponential function for which every ๐‘ฆ-value is multiplied by ๐‘ when ๐‘ฅ increases by ฮ”๐‘ฅ. We are asked to find ๐‘ when ฮ”๐‘ฅ=1.

Keep in mind that ฮ”๐‘ฅ=1 means a change of 1 in the value of ๐‘ฅ. Since we have already established that the graph passes through the point (0,10), it would be easiest for us to investigate what happens when the ๐‘ฅ-coordinate of a point that the graph passes through increases from 0 to 1. This is a change of 1 in the ๐‘ฅ-value.

We can see that the graph passes through the point (1,20), so when the value of ๐‘ฅ changes from 0 to 1, the value of ๐‘ฆ changes from 10 to 20. Remember that we were told that every ๐‘ฆ-value is multiplied by ๐‘ when ๐‘ฅ increases by ฮ”๐‘ฅ. Thus, to find the value of ๐‘ when ฮ”๐‘ฅ=1โˆ’0=1, we must determine what number 10 is multiplied by to get 20. We can do this by dividing 20 by 10: 2010=2.

This tells us that ๐‘ is 2 when ฮ”๐‘ฅ=1.

Part 3

Now that we know the value of ๐‘ for a specific value of ฮ”๐‘ฅ, we can determine the equation that describes the graph in the form ๐‘ฆ=๐‘Ž๐‘๏‘๏ซ๏‘. First, we can substitute 2 for ๐‘ and 1 for ฮ”๐‘ฅ to get ๐‘ฆ=๐‘Žโ‹…2=๐‘Žโ‹…2.๏‘๏Ž ๏—

Next, to find the value of ๐‘Ž, we can begin by substituting the coordinates of one of the points that the graph passes through into the equation ๐‘ฆ=๐‘Žโ‹…2๏—. Here, we will substitute the coordinates of the point (0,10) into the equation, which gives us 10=๐‘Žโ‹…2.๏Šฆ

We can then simplify to get 10=๐‘Žโ‹…1=๐‘Ž.

Therefore, we know that the value of ๐‘Ž is 10. Recall that we also know that ๐‘ is 2 when ฮ”๐‘ฅ=1, so we now have values for ๐‘Ž, ๐‘, and ฮ”๐‘ฅ that we can substitute into ๐‘ฆ=๐‘Ž๐‘๏‘๏ซ๏‘to find the equation of the graph.

Substituting shows us that the equation that describes the graph in the form ๐‘ฆ=๐‘Ž๐‘๏‘๏ซ๏‘ is ๐‘ฆ=10โ‹…2๏‘๏Ž , or ๐‘ฆ=10โ‹…2๏—.

Next, we will work on a problem in which we must determine the value of an expression by evaluating an exponential function.

Example 5: Evaluating Exponential Functions

Given that ๐‘“(๐‘ฅ)=4๏—, determine the value of ๐‘“(๐‘ฅ)๐‘“(๐‘ฅโˆ’1)โˆ’๐‘“(๐‘ฅโˆ’1)๐‘“(๐‘ฅ).

Answer

In this problem, we are asked to find the value of an expression that represents the difference of the two fractions ๐‘“(๐‘ฅ)๐‘“(๐‘ฅโˆ’1) and ๐‘“(๐‘ฅโˆ’1)๐‘“(๐‘ฅ). However, we are only told that ๐‘“(๐‘ฅ)=4๏— and are not given the value of ๐‘ฅ. This means that in order to solve the problem, we must determine the relationship between ๐‘“(๐‘ฅ) and ๐‘“(๐‘ฅโˆ’1). To do this, letโ€™s begin by evaluating the function ๐‘“(๐‘ฅ)=4๏— for some different values of ๐‘ฅ.

๐‘ฅ1234
๐‘“(๐‘ฅ)4=4๏Šง4=4โ‹…4=16๏Šจ4=4โ‹…4โ‹…4=64๏Šฉ4=4โ‹…4โ‹…4โ‹…4=256๏Šช

We can see that for each increase of 1 in the value of ๐‘ฅ, the previous value of ๐‘“(๐‘ฅ) is multiplied by 4. That is,

  • the value of ๐‘“(2) is ๐‘“(1)โ‹…4,
  • the value of ๐‘“(3) is ๐‘“(2)โ‹…4,
  • the value of ๐‘“(4) is ๐‘“(3)โ‹…4.

In general, we can say that the value of ๐‘“(๐‘ฅ) is equal to the value of ๐‘“(๐‘ฅโˆ’1) times 4. This relationship between ๐‘“(๐‘ฅ) and ๐‘“(๐‘ฅโˆ’1) will be true for any real number value of ๐‘ฅ.

Since we know that ๐‘“(๐‘ฅ)=4๐‘“(๐‘ฅโˆ’1), we also know that ๐‘“(๐‘ฅโˆ’1)=14๐‘“(๐‘ฅ). With this knowledge, we can be certain that the value of the fraction ๐‘“(๐‘ฅ)๐‘“(๐‘ฅโˆ’1) is 4 and that the value of the fraction ๐‘“(๐‘ฅโˆ’1)๐‘“(๐‘ฅ) is 14.

Substituting 4 for ๐‘“(๐‘ฅ)๐‘“(๐‘ฅโˆ’1) and 14 for ๐‘“(๐‘ฅโˆ’1)๐‘“(๐‘ฅ) into the expression given in the problem and simplifying, we get ๐‘“(๐‘ฅ)๐‘“(๐‘ฅโˆ’1)โˆ’๐‘“(๐‘ฅโˆ’1)๐‘“(๐‘ฅ)=4โˆ’14=164โˆ’14=154.

Thus, given that ๐‘“(๐‘ฅ)=4๏—, we know that the value of ๐‘“(๐‘ฅ)๐‘“(๐‘ฅโˆ’1)โˆ’๐‘“(๐‘ฅโˆ’1)๐‘“(๐‘ฅ) must be 154.

Note

An alternative solution method involves substituting for both ๐‘“(๐‘ฅ) and ๐‘“(๐‘ฅโ€“1) in the expression given in the problem and then simplifying. Since ๐‘“(๐‘ฅ)=4๏—, we know that ๐‘“(๐‘ฅโˆ’1)=4๏—๏Šฑ๏Šง. Substituting 4๏— for ๐‘“(๐‘ฅ) and 4๏—โ€“๏Šง for ๐‘“(๐‘ฅโ€“1), we get 44โˆ’44.๏—๏—๏Šฑ๏Šง๏—๏Šฑ๏Šง๏—

Now, simplifying gives us 4โˆ’4=4โˆ’4=4โˆ’4=4โˆ’14=164โˆ’14=154.๏—๏Šฑ(๏—๏Šฑ๏Šง)๏—๏Šฑ๏Šง๏Šฑ๏—๏—๏Šฑ๏—๏Šฐ๏Šง๏—๏Šฑ๏Šง๏Šฑ๏—๏Šง๏Šฑ๏Šง

This is the same value of ๐‘“(๐‘ฅ)๐‘“(๐‘ฅโˆ’1)โˆ’๐‘“(๐‘ฅโˆ’1)๐‘“(๐‘ฅ) that we arrived at previously.

Check

We may be able to identify any possible mistakes by evaluating ๐‘“(๐‘ฅ) and ๐‘“(๐‘ฅโˆ’1) at a specific value of ๐‘ฅ. In this case, letโ€™s use ๐‘ฅ=2. This gives us ๐‘“(2)=4=16๐‘“(2โˆ’1)=๐‘“(1)=4=4,๏Šจ๏Šงand so the value of our expression becomes ๐‘“(2)๐‘“(2โˆ’1)โˆ’๐‘“(2โˆ’1)๐‘“(2)=164โˆ’416=4โˆ’14.

We have already determined that the difference of 4 and 14 is 154, so we know that 154 is the value of the expression ๐‘“(๐‘ฅ)๐‘“(๐‘ฅโˆ’1)โˆ’๐‘“(๐‘ฅโˆ’1)๐‘“(๐‘ฅ) when ๐‘ฅ=2. This is what we would expect, since the value of the expression should be 154 for any value of ๐‘ฅ.

Note that, in addition to the quotient rule, we used the negative exponent rule in our alternative solution. This is another one of the properties of exponents.

Definition: Negative Exponent Rule

The negative exponent rule states that any base raised to a negative power is equal to 1 over the base raised to the exponentโ€™s additive inverse. That is, ๐‘=1๐‘,๏Šฑ๏Š๏Š where ๐‘ is the base and ๐‘› and โˆ’๐‘› are the exponents.

For example, we would use the negative exponent rule to find that 6=16๏Šฑ๏Šจ๏Šจ.

Finally, we will determine the values of an exponential functionโ€™s base that result in the function decreasing without being told the form of the function.

Example 6: Discussing the Monotonicity of Exponential Functions

Consider an exponential function with base ๐‘Ž. For which values of ๐‘Ž is the function decreasing?

Answer

Letโ€™s begin by supposing that we have the exponential function ๐‘“(๐‘ฅ)=๐‘Ž๏—. This function is a monotonic function. That is, it is a function that is always increasing or always decreasing. For this reason, if we can determine that ๐‘“(๐‘ฅ)=๐‘Ž๏— is decreasing over some interval of ๐‘ฅ, then we will know that it is decreasing over all real numbers ๐‘ฅ.

With this in mind, letโ€™s consider the function over the interval [1,4]. Finding ๐‘“(๐‘ฅ) for the integer values of ๐‘ฅ in this interval, we get the following table of values.

๐‘ฅ1234
๐‘“(๐‘ฅ)๐‘Ž=๐‘Ž๏Šง๐‘Ž=๐‘Žโ‹…๐‘Ž๏Šจ๐‘Ž=๐‘Žโ‹…๐‘Žโ‹…๐‘Ž๏Šฉ๐‘Ž=๐‘Žโ‹…๐‘Žโ‹…๐‘Žโ‹…๐‘Ž๏Šช

We can see that for each increase of 1 in the value of ๐‘ฅ, the previous value of ๐‘“(๐‘ฅ) is multiplied by ๐‘Ž. That is,

  • the value of ๐‘“(2) is ๐‘“(1)โ‹…๐‘Ž,
  • the value of ๐‘“(3) is ๐‘“(2)โ‹…๐‘Ž,
  • the value of ๐‘“(4) is ๐‘“(3)โ‹…๐‘Ž.

Remember that for any exponential function, the base must be a positive real number not equal to 1. Therefore, to determine the values of ๐‘Ž for which the function is decreasing, we must ask ourselves, โ€œFor what positive real number values of a not equal to 1 is the value of ๐‘“(๐‘ฅ)โ‹…๐‘Ž less than the value of ๐‘“(๐‘ฅ)?โ€

First letโ€™s consider a value of ๐‘Ž greater than 1. For example, if ๐‘Ž=3, then evaluating the function for the integer values of ๐‘ฅ in the interval [1,4] gives

  • ๐‘“(1)=3=3๏Šง,
  • ๐‘“(2)=3=3โ‹…3=9๏Šจ,
  • ๐‘“(3)=3=3โ‹…3โ‹…3=27๏Šฉ,
  • ๐‘“(4)=3=3โ‹…3โ‹…3โ‹…3=81๏Šช.

This shows that if ๐‘Ž=3, then the value of ๐‘“(๐‘ฅ)โ‹…๐‘Ž is greater than the value of ๐‘“(๐‘ฅ). This will always be true when ๐‘Ž>1. In this case, ๐‘Ž is referred to as the growth factor.

Now letโ€™s consider a value of ๐‘Ž less than 1. Since ๐‘Ž must be positive, its value will also be greater than 0. For example, if ๐‘Ž=13, then evaluating the function for the integer values of ๐‘ฅ in the interval [1,4] gives

  • ๐‘“(1)=๏€ผ13๏ˆ=13๏Šง,
  • ๐‘“(2)=๏€ผ13๏ˆ=13โ‹…13=19๏Šจ,
  • ๐‘“(3)=๏€ผ13๏ˆ=13โ‹…13โ‹…13=127๏Šฉ,
  • ๐‘“(4)=๏€ผ13๏ˆ=13โ‹…13โ‹…13โ‹…13=181๏Šช.

This shows that if ๐‘Ž=13, then the value of ๐‘“(๐‘ฅ)โ‹…๐‘Ž is less than the value of ๐‘“(๐‘ฅ). This will always be true when 0<๐‘Ž<1. In this case, ๐‘Ž is referred to as the decay factor.

Since the value of ๐‘“(๐‘ฅ)โ‹…๐‘Ž is less than the value of ๐‘“(๐‘ฅ) in the interval [1,4] when 0<๐‘Ž<1, we know that ๐‘“(๐‘ฅ) is decreasing in this interval. Also, because of the functionโ€™s monotonicity, we know that if the function is decreasing in this interval, then it is decreasing over all real numbers.

Thus, the values of a for which the function is decreasing are 0<๐‘Ž<1.

Note

Although we chose ๐‘“(๐‘ฅ)=๐‘Ž๏— as our function, we could have chosen a transformed exponential function and arrived at the same result.

While answering the previous question, we saw that if an exponential function increases or decreases over a certain interval, then it does the same over all real numbers. This is because of its monotonicity.

Property: Monotonicity of Exponential Functions

An exponential function is a monotonic function, or a function that always increases or always decreases. If the base is greater than 1, it is called the growth factor, and the function will always increase. If the base is greater than 0 and less than 1, it is called the decay factor, and the function will always decrease.

Now letโ€™s finish by recapping some key points.

Key Points

  • An exponential functionโ€™s rule has the form ๐‘“(๐‘ฅ)=๐‘๏—, where the constant ๐‘ is the base and the variable ๐‘ฅ is the exponent.
  • The base of an exponential function must be a positive real number not equal to 1.
  • The quotient rule states that when dividing exponential expressions with the same base, we keep the base and find the difference of the exponents. That is ๐‘๐‘=๐‘,๏‰๏Š๏‰๏Šฑ๏Š where ๐‘ is the base and ๐‘š and ๐‘› are the exponents.
  • The zero exponent rule states that any base raised to the power 0 is equal to 1. That is, ๐‘=1,๏Šฆ where ๐‘ is the base.
  • The negative exponent rule states that any base raised to a negative power is equal to 1 over the base raised to the exponentโ€™s additive inverse. That is, ๐‘=1๐‘,๏Šฑ๏Š๏Š where ๐‘ is the base and ๐‘› and โˆ’๐‘› are the exponents.
  • An exponential function is a monotonic function, or a function that always increases or always decreases.
  • If the base of an exponential function is greater than 1, it is called the growth factor, and the function will always increase. If the base is greater than 0 and less than 1, it is called the decay factor, and the function will always decrease.

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