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Learning objectives

  • Evaluate exponential expressions
  • Identify the domain and range of exponential functions
  • Graph exponential functions by hand and using a graphing utility
  • Solve basic exponential equations

Introduction

In this lesson you will learn about exponential functions, a family of functions we have not studied in chapter 1 or chapter 2. In terms of the form of the equation, exponential functions are different from the other function families because the variable x is in the exponent. For example, the functions f(x) = 2x and g(x) = 100(2)5x are exponential functions. This kind of function can be used to model real situations, such as population growth, compound interest, or the decay of radioactive materials. In this lesson we will look at basic examples of these functions, and we will graph and solve exponential equations. This introduction to exponential functions will prepare you to study applications of exponential functions later in this chapter.

Evaluating Exponential Functions

Consider the function f(x) = 2x. When we input a value for x, we find the function value by raising 2 to the exponent of x. For example, if x = 3, we have f(3) = 23 = 8. If we choose larger values of x, we will get larger function values, as the function values will be larger powers of 2. For example, f(10) = 210 = 1,024.

Now let’s consider smaller x values. If x = 0, we have f(0) = 20 = 1. If x = -3, we have f(-3)=2^{-3}=\left( \tfrac{1}{2} \right)^{3}=\tfrac{1}{8}. If we choose smaller and smaller x values, the function values will be smaller and smaller fractions. For example, if x = -10, we have f(-10)=2^{-10}=\left( \tfrac{1}{2} \right)^{10}=\tfrac{1}{1024}. Notice that none of the x values we choose will result in a function value of 0. (This is the case because the numerator of the fraction will always be 1.) This tells us that while the domain of this function is the set of all real numbers, the range is limited to the set of positive real numbers. In the following example, you will examine the values of a similar function.

Example 1: For the function g(x) = 3x, find g(2), g(4), g(0), g(-2), g(-4).

Solution:

g(2) = 3^2 = 9

g(4) = 3^4 = 81

g(0) = 3^0 = 1

g(-2) = 3^{-2} = \frac{1} {3^2} = \frac{1} {9}

g(-4) = 3^{-4} = \frac{1} {3^4} = \frac{1} {81}

The values of the function g(x) = 3x behave much like those of f(x) = 2x: if we choose larger values, we get larger and larger function values. If x = 0, the function value is 1. And, if we choose smaller and smaller x values, the function values will be smaller and smaller fractions. Also, the range of g(x) is limited to positive values.

In general, if we have a function of the form f(x) = ax, where a is a positive real number, the domain of the function is the set of all real numbers, and the range is limited to the set of positive real numbers. This restricted domain will result in a specific shape of the graph.

Graphing basic exponential functions

Let’s now consider the graph of f(x) = 2x. Above we found several function values, and we began to analyze the function in terms of large and small values of x. The graph below shows this function, with several points marked in blue.

Notice that as x approaches \infty, the function grows without bound. That is, \lim_{x \to \infty} (2^x) = \infty. However, if x approaches -\infty, the function values get closer and close to 0. That is, \lim_{x \to -\infty} (2^x) = 0. Therefore the function is asymptotic to the x-axis. This is the graphical result of the fact that the range of the function is limited to positive y values. Now let’s consider the graph of g(x) = 3x and h(x) = 4x.

Example 2: Use a graphing utility to graph f(x) = 2x, g(x) = 3x and h(x) = 4x. How are the graphs the same, and how are they different?

Solution: f(x) = 2x, g(x) = 3x and h(x) = 4x are shown together below.

The graphs of the three functions have the same overall shape: they have the same end behavior, and they all contain the point (0, 1). The difference lies in their rate of growth. Notice that for positive x values, h(x) = 4x grows the fastest and f(x) = 2x grows the slowest. The function values for h(x) = 4x are highest and the function values for f(x) = 2x are the lowest for any given value of x. For negative x values, the relationship changes: f(x) = 2x has the highest function values of the three functions.

Now that we have examined these three parent graphs, we will graph using shifts, reflections, stretches and compressions.

Graphing exponential functions using transformations.

Above we graphed the function f(x) = 2x. Now let’s consider a related function: g(x) = 2x + 3. Every function value will be a power of 2, plus 3. The table below shows several values for the function:

x g(x) = 2^x + 3
-2 2^{-2} + 3 = \frac{1} {4} + 3 = 3 \frac{1} {4}
-1 2^{-1} + 3 = \frac{1} {2} + 3 = 3 \frac{1} {2}
0 2^{0} + 3 = 1 + 3 = 4
1 2^{1} + 3 = 2 + 3 = 5
2 2^{2} + 3 = 4 + 3 = 7
3 2^{3} + 3 = 8 + 3 = 11

The function values follow the same kind of pattern as the values for f(x) = 2x. However, because every function value is 3 more than a power of 2, the horizontal asymptote of the function is the line y = 3. The graph of this function and the horizontal asymptote are shown below.

From your study of transformation of functions in chapter 1, you should recognize the graph of g(x) = 2x + 3 as a vertical shift of the graph of f(x) = 2x. In general, we can produce a graph of an exponential function with base 2 if we analyze the equation of the function in terms of transformations. The table below summarizes the different kinds of transformations of f(x) = 2x. The issue of stretching will be discussed further below the table.

Equation Relationship to f(x)=2x Range
g(x)=\frac{2^x}{2^a}=2^{x-a}, \text{for } a>0 Obtain a graph of g by shifting the graph of f a units to the right. y > 0
g(x)=2^{a} \cdot 2^{x}=2^{a+x}, \text{for } a>0 Obtain a graph of g by shifting the graph of f a units to the left. y > 0
\,\! g(x)=2^x+a, \text{for } a>0 Obtain a graph of g by shifting the graph of f up a units. y > a
\,\! g(x)=2^x-a, \text{for } a>0 Obtain a graph of g by shifting the graph of f down a units. y > a
\,\! g(x)=a(2^x), \text{for } a>0 Obtain a graph of g by vertically stretching the graph of f by a factor of a. y > 0
\,\! g(x)=2^{ax}, \text{for } a>0 Obtain a graph of g by horizontally compressing the graph of f by a factor of a. y > 0
\,\! g(x)=-2^x Obtain a graph of g by reflecting the graph of f over the x-axis. y > 0
\,\! g(x)=2^{-x} Obtain a graph of g by reflecting the graph of f over the y-axis. y > 0

As was discussed in chapter 1, a stretched graph can also be seen as a compressed graph. This is not the case for exponential functions because of the x in the exponent. Consider the function s(x) = 2(2x) and c(x) = 23x. The first function represents a vertical stretch of f(x) = 2x by a factor of 2. The second function represents a compression of f(x) = 2x by a factor of 3. The function c(x) is actually the same as another parent function: c(x) = 23x = (23)x = 8x. The function s(x) is actually the same as a shift of f(x) = 2x: s(x) = 2(2x) = 21 × 2x = 2x + 1. The graphs of s and c are shown below. Notice that the graph of c has a y-intercept of 1, while the graph of s has a y-intercept of 2:

Example 3: Use transformations to graph the functions (a) a(x) = 3x + 2 and (b) b(x) = -3x + 4

Solution:

a. a(x) = 3x + 2

This graph represents a shift of y = 3x two units to the left. The graph below shows this relationship between the graphs of these two functions:

b. b(x) = -3x + 4

This graph represents a reflection over the y-axis and a vertical shift of 4 units. You can produce a graph of b(x) using three steps: sketch y = 3x, reflect the graph over the x-axis, and then shift the graph up 4 units. The graph below shows this process:

While you can always quickly create a graph using a graphing utility, using transformations will allow you to sketch a graph relatively quickly on your own. If we start with a parent function such as y=3x, you can quickly plot several points: (0, 1), (2, 9), (-1, 1/3), etc. Then you can transform the graph, as we did in the previous example.

Notice that when we sketch a graph, we choose x values, and then use the equation to find y values. But what if we wanted to find an x value, given a y value? This requires solving exponential equations.

Solving exponential equations

Solving an exponential equation means determining the value of x for a given function value. For example, if we have the equation 2x = 8, the solution to the equation is the value of x that makes the equation a true statement. Here, the solution is x = 3, as 23 = 8.

Consider a slightly more complicated equation 3 (2x + 1) = 24. We can solve this equation by writing both sides of the equation as a power of 2:

3(2^{x + 1}) = 24
\frac{3(2^{x + 1})} {3} = \frac{24} {3}
2^{x + 1} = 8
2^{x + 1} = 2^3

To solve the equation now, recall a property of exponents: if bx = by, then x = y. That is, if two powers of the same base are equal, the exponents must be equal. This property tells us how to solve:

2^{x + 1} = 2^3
\Rightarrow x + 1 = 3
 \,\! x = 2

Example 4: Solve the equation 56x + 10 = 25x - 1

Solution: Use the same technique as shown above:

56x + 10 = 25x - 1
56x + 10 = (52)x - 1
56x + 10 = 52x - 2
\Rightarrow 6x + 10 = 2x - 2
4x + 10 = -2
4x = -12
x = -3

In both of the examples of solving equations, it was possible to solve because we could write both sides of the equations as a power of the same exponent. But what if that is not possible?

Consider for example the equation 3x = 12. If you try to figure out the value of x by considering powers of 3, you will quickly discover that the solution is not a whole number. Later in the chapter we will study techniques for solving more complicated exponential equations. Here we will solve such equations using graphs.

Consider the function y = 3x. We can find the solution to the equation 3x = 12 by finding the intersection of y = 3x and the horizontal line y = 12. Using a graphing calculator’s intersection capability, you should find that the solution is approximately x = 2.26.

Example 5: Use a graphing utility to solve each equation:

a. 23x - 1 = 7 b. 6-4x = 28x - 5

Solution:

a. 23x - 1 = 7

Graph the function y = 23x - 1 and find the point where the graph intersects the horizontal line y = 7. The solution is x ≈ 1.27

b. 6-4x = 28x - 5

Graph the functions y = 6-4x and y = 28x - 5 and find their intersection point.

The solution is approximately x ≈ 0.27. (Your graphing calculator should show 9 digits: 0.272630365.)

In the examples we have considered so far, the bases of the functions have been positive integers. Now we will examine a sub-family of exponential functions with a special base: the number e.

The number e and the function y = ex

In your previous studies of math, you have likely encountered the number π. The number e is much like π. First, both are irrational numbers: they cannot be expressed as fractions. Second, both numbers are transcendental: they are not the solution of any polynomial with rational coefficients.

Like π, mathematicians found e to be a natural constant in the world. One way to “discover” e is to consider the function f(x) = \left (1 + \frac{1} {x}\right )^x. The graph of this function is shown below.

Notice that as x approaches infinity, the graph of the function approaches a horizontal asymptote around y = 2.7. If you examine several function values, you will see that the limit is not exactly 2.7:

x y
0 (not defined)
1 2
2 2.25
5 2.48832
10 2.5937424601
50 2.69158802907
100 2.70481382942
1000 2.71692393224
5000 2.7180100501
10,000 2.71814592683
50,000 2.7181825464614

Around x = 100, the function values pass 2.7, but they will never reach 2.8. The limit of the function as x approaches infinity is the constant e. The value of e is approximately 2.71828182845904523536. Again, like π, we have to approximate the value of e because it is irrational.

The number e is used as the base of functions that can be used to model situations that involve growth or decay. For example, as you will learn later in the chapter, one method of calculating interest on a bank account or investment uses this number. Here we will examine the function y = ex in order to verify that its graph is similar to the other exponential functions we have graphed.

The graph below shows y = ex, along with y = 2x and y = 3x.

The graph of y = ex (in green) has the same shape as the graphs of the other exponential functions. It sits in between the graphs of the other two functions, and notice that the graph is closer to y = 3x than to y = 2x. All three graphs have the same y-intercept: (0, 1). Thus the graph of this function is clearly a member of the same family, even though the base of the function is an irrational number.

Lesson Summary

This lesson has introduced the family of exponential functions. We have examined values of functions, towards understanding the behavior of graphs. In general, exponential functions have a horizontal asymptote, though one end of the function increases (or decreases, if it is a reflection) without bound.

In this lesson we have graphed these functions, solved certain exponential equations using our knowledge of exponents, and solved more complicated equations using graphing utilities. We have also examined the function y = ex, which is a special member of the exponential family. In the coming lessons you will continue to learn about exponential functions, including the inverses of these functions, applications of these functions, and solving more complicated exponential equations using algebraic techniques.

Points to Consider

  1. Why do exponential functions have horizontal asymptotes and not vertical asymptotes?
  2. What would the graph of the inverse of an exponential function look like? What would its domain and range be?
  3. How could you solve or approximate a solution to an exponential equation without using a graphing calculator?

Review Questions

  1. For the function f(x) = 23x - 1, find f(0), f(2), and f(-2).
  2. Graph the functions f(x) = 3x and g(x) = 3x + 5 -1. State the domain and range of each function.
  3. Graph the functions a(x) = 4x and b(x) = 4-x. State the domain and range of each function.
  4. Graph the function h(x) = -2x - 1 using transformations. How is h related to y = 2x?
  5. Solve the equation: 52x + 1 = 253x
  6. Solve the equation: 4x2 + 1 = 16x
  7. Use a graph to find an approximate solution to the equation 3x = 14
  8. Use a graph to find an approximate solution to the equation 2-x = 72x + 9
  9. Sketch a graph of the function f(x) = 3x and its inverse. (Hint: You can graph the inverse by reflecting a function across the line y=x.) Is f one-to-one?
  10. Consider the following situation: you inherited a collection of 125 stamps from a relative. You decided to continue to build the collection, and you vowed to double the size of the collection every year. a. Write an exponential function to model the situation. (The input of the function is the number of years since you began building the collection, and the output is the size of the collection.) b. Use your model to determine how long it will take to have a collection of 10,000 stamps.

Review Answers

  1. f(0)=1/2 , f(2)= 32, f(-2)= 1/128
  2. The domain if both functions is the set of all real numbers. The range of f is the set of all real numbers ≥ 0. The range of g is the set of all real numbers ≥ -1
  3. The domain of both functions is the set of all real numbers. The range of both functions is the set of all real numbers ≥ 0.
  4. The function h represents a reflection over the y axis, and a horizontal shift 1 unit to the right.
  5. 52x + 1 = 253x 52x + 1 = 56x \Rightarrow 2x+ 1 = 6x 4x = 1 x = 1/4
  6. 4x2 + 1 = 16x 4x2 + 1 = 42x  \Rightarrowx2 + 1 = 2x x2 - 2x + 1 = 0 (x - 1) (x - 1) = 0 x = 1
  7. x ≈ 2.4
  8. x ≈ -3.8
  9. f is a one-to-one function.
  10. a. S(t) = 125 (2t) b. About 6.35 years

Vocabulary

e
The number e is a transcendental number, often referred to as Euler’s constant. Several mathematicians are credited with early work on e. Euler was the first to use this letter to represent the constant. The value of e is approximately 2.71828. The exact value is \lim_{x \to \infty} \left (1 + \frac{1} {x}\right )^x.
Exponential Function
An exponential function is a function for which the input variable x is in the exponent of some base b, where b is a real number.
Irrational number
An irrational number is a number that cannot be expressed as a fraction of two integers.
Transcendental number
A transcendental number is a number that is not a solution to any non-zero polynomial with rational roots.
π
The number π is a transcendental number. It is the ratio of the circumference to the diameter in any circle.

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