7.2 Exponential Functions And Their Graphs

Exponential Functions

At this point in our study of algebra we begin to look at transcendental functions or functions that seem to “transcend” algebra. We have studied functions with variable bases and constant exponents such as x2 or y−3. In this section we explore functions with a constant base and variable exponents. Given a real number b>0 where b≠1 an exponential functionAny function with a definition of the form f(x)=bx where b>0 and b≠1. has the form,

f(x)=bx     Exponential Function

For example, if the base b is equal to 2, then we have the exponential function defined by f(x)=2x. Here we can see the exponent is the variable. Up to this point, rational exponents have been defined but irrational exponents have not. Consider 27, where the exponent is an irrational number in the range,

2.64<7<2.65

We can use these bounds to estimate 27,

22.64<27<22.656.23<27<6.28

Using rational exponents in this manner, an approximation of 27 can be obtained to any level of accuracy. On a calculator,

2^7≈6.26

Therefore the domain of any exponential function consists of all real numbers (−∞,∞). Choose some values for x and then determine the corresponding y-values.

xyf(x)=2xSolutions−214y=2−2=122=14(−2, 14)−112y=2−1=121=12(−1, 12)01y=20=1(0, 1)12y=21=2(1, 2)24y=22=4(2, 4)76.26y=27≈6.26(2.65, 6.26)

Because exponents are defined for any real number we can sketch the graph using a continuous curve through these given points:

It is important to point out that as x approaches negative infinity, the results become very small but never actually attain zero. For example,

f(−5)=2−5=125≈0.03125f(−10)=2−10=1210≈0.0009766f(−15)=2−15=12−15≈.00003052

This describes a horizontal asymptote at y=0, the x-axis, and defines a lower bound for the range of the function: (0,∞).

The base b of an exponential function affects the rate at which it grows. Below we have graphed y=2x, y=3x, and y=10x on the same set of axes.

Note that all of these exponential functions have the same y-intercept, namely (0,1). This is because f(0)=b0=1 for any function defined using the form f(x)=bx. As the functions are read from left to right, they are interpreted as increasing or growing exponentially. Furthermore, any exponential function of this form will have a domain that consists of all real numbers (−∞,∞) and a range that consists of positive values (0,∞) bounded by a horizontal asymptote at y=0.

Example 1

Sketch the graph and determine the domain and range: f(x)=10x+5.

Solution:

The base 10 is used often, most notably with scientific notation. Hence, 10 is called the common base. In fact, the exponential function y=10x is so important that you will find a button 10x dedicated to it on most modern scientific calculators. In this example, we will sketch the basic graph y=10x and then shift it up 5 units.

Note that the horizontal asymptote of the basic graph y=10x was shifted up 5 units to y=5 (shown dashed). Take a minute to evaluate a few values of x with your calculator and convince yourself that the result will never be less than 5.

Answer:

Domain: (−∞,∞); Range: (5,∞)

Next consider exponential functions with fractional bases 0<b<1. For example, f(x)=(12)x is an exponential function with base b=12.

xyf(x)=(12)xSolutions−24f(12)=(12)−2=1−22−2=2212=4(−2, 4)−12f(12)=(12)−1=1−12−1=2111=2(−1, 2)01f(12)=(12)0=1(0, 1)112f(12)=(12)1=12(1, 12)214f(12)=(12)2=14(2, 14)

Plotting points we have,

Reading the graph from left to right, it is interpreted as decreasing exponentially. The base affects the rate at which the exponential function decreases or decays. Below we have graphed y=(12)x, y=(13)x, and y=(110)x on the same set of axes.

Recall that x−1=1x and so we can express exponential functions with fractional bases using negative exponents. For example,

g(x)=(12)x=1x2x=12x=2−x.

Furthermore, given that f(x)=2x we can see g(x)=f(−x)=2−x and can consider g to be a reflection of f about the y-axis.

In summary, given b>0

And for both cases,

Domain:(−∞,∞)Range:​(0,∞)y-intercept:(0,1)Asymptote:​​​y=0

Furthermore, note that the graphs pass the horizontal line test and thus exponential functions are one-to-one. We use these basic graphs, along with the transformations, to sketch the graphs of exponential functions.

Example 2

Sketch the graph and determine the domain and range: f(x)=5−x−10.

Solution:

Begin with the basic graph y=5−x and shift it down 10 units.

The y-intercept is (0,−9) and the horizontal asymptote is y=−10.

Answer:

Domain: (−∞,∞); Range: (−10,∞)

Note: Finding the x-intercept of the graph in the previous example is left for a later section in this chapter. For now, we are more concerned with the general shape of exponential functions.

Example 3

Sketch the graph and determine the domain and range: g(x)=−2x−3.

Solution:

Begin with the basic graph y=2x and identify the transformations.

y=2x  Basic graphy=−2x  Reflection about the x-axisy=−2x−3  Shift right 3 units

Note that the horizontal asymptote remains the same for all of the transformations. To finish we usually want to include the y-intercept. Remember that to find the y-intercept we set x=0.

g(0)=−20−3=−2−3=−123=−18

Therefore the y-intercept is (0, −18).

Answer:

Domain: (−∞,∞); Range: (−∞,0)

Try this! Sketch the graph and determine the domain and range: f(x)=2x−1+3.

Answer:

Domain: (−∞,∞); Range: (3,∞)

(click to see video)