In your previous math courses, you have experienced problems concerning exponential growth and decay. Business and biology have problems that use an exponential growth model, and the physical sciences (Chemistry and Physics) have problems that use an exponential decay model. In this set of notes, I will review the properties of the exponential function. To start this process, I will state the definition of an exponential function.

DEFINITION:

Let a be a positive real number other than 1. The function f (x) = a ^x is the exponential function with base a.

a > 1

(See figure 1)

figure 1

DOMAIN:

(- ¥ , ¥ )

RANGE:

(0, ¥ )

INCREASING

0 < a < 1

(See figure 2)

figure 2

DOMAIN:

(- ¥ , ¥ )

RANGE:

(0, ¥ )

DECREASING

 Now it is time to look at a couple of examples on how to determine the domain, range, and intercepts of an exponential function.

EXAMPLE 1:

Find the domain, range, and intercepts for y = -2 ^{- x} - 1.

SOLUTION:

The domain is the easiest trait to determine for an exponential function. It is always (- ¥ , ¥ ).

The easiest way to determine the range for this function is to look at a graph of it. (See figure 3) To determine the range, you will read the graph from bottom to top. Notice that the graph starts at a - ¥ and approaches -1.

figure 3

(Algebraically, as x goes to ¥ , -2 ^{- x} goes to 0, so y goes to -1.) Therefore, the range for this function is (-¥ , -1).

Now we must determine the intercepts. From the graph, we can see that this function only has one intercept, and it is the y-intercept. To find the y-intercept algebraically, let x = 0 and solve for y.

When x = 0, y = -2 ⁰ - 1 = -1 - 1 = -2.

The graph shows us that this function only has one intercept, but what if we did not have a graph to look at? We would then have to determine if there was an x-intercept algebraically by letting y = 0 and solving for x.

When y = 0, -2 ^{- x} - 1 = 0 ® -2 ^{- x} = 1 (This cannot happen. A negative number cannot equal a positive number!)

So in summary, the domain for this function is (- ¥ , ¥ ). The range for this function is (- ¥ , -1), and the y-intercept is (0, -2).

EXAMPLE 2:

Find the domain, range, and intercepts for y = -2 ^x +3.

SOLUTION:

The domain is the easiest trait to determine for an exponential function. It is always (- ¥ , ¥ ).

The easiest way to determine the range for this function is to look at a graph of it. (See figure 4) To determine the range, you will read the graph from bottom to top. Notice that the graph starts at a - ¥ and approaches 3.

figure 4

(Algebraically, as x goes to ¥ , -2 ^x goes to 0, so y goes to 3.) The range for this function is (-¥ , 3).

Now we must determine the intercepts. From the graph, we can see that this function has two intercepts, and we will find the y-intercept first. To find the y-intercept algebraically, let x = 0 and solve for y.

When x = 0, y = -2 ⁰ + 3 = -1 + 3 = 2.

When y = 0, -2 ^x + 3 = 0 ® 2 ^x = 3 ® x ln 2 = ln 3 ® x = (ln 3)/(ln 2).

So in summary, the domain of this function is (- ¥ , ¥ ). The range of this function is (- ¥ , 3). The x-intercept is ((ln 3)/(ln 2), 0) and the y-intercept is (0, 2).

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

1.

a ^x · a ^y = a ^{x + y}

2.

3.

(a ^x) ^y = (a ^y) ^x = a ^{x y}

4.

a ^x · b ^x = (ab) ^x

5.

EXAMPLE 3:

Assume that the graph of the exponential function f (x) = k(a ^x) passes through the points (-1, -3) and (1, -1.5). Find the values of k and a. 

SOLUTION: Since we are given two points in which the graph of the equation passes through, we can then use those two points to define two equations that have two unknowns. Then we will solve this system for the two unknowns.

At the point (-1, -3), the equation will be -3 = ka ^{- 1}.

At the point (1, -1.5), the equation will be -1.5 = ka ¹.

I will solve the first equation for k in terms of a.

Now I will plug this result into the second equation. I will solve that new equation for a.

Now I will plug the value for a into the equation -3a = k.

These are the two values that we will need to put into the equation. When we put these two values into the equation, then we should be able to determine if the graph passes through those points.

This concludes the review of the topic exponential functions. Please work through these examples and the homework problems in the text. If you have any questions on any of the material, please feel free to contact me.