Exponential Functions

The definition of exponential functions are discussed using graphs and values. The properties such as domain, range, horizontal asymptotes, x and y intercepts are also presented. The conditions under which an exponential function increases or decreases are also investigated.

Definition of the Exponential Function

The basic exponential function is defined by \[ f(x) = B^{x} \]
where \( B \) is the base of the exponential such that \( B \gt 0 \) and \( B \ne 1 \).
The domain of the exponential function \( f \), defined above, is the set of all real numbers.

Example 1: Table of values and graphs of exponential functions with base greater than 1
A table of values and the graphs of the exponential functions \( 2^x \), \( 4^x \) and \( 7^x\) are shown below

\( x \) \( 2^x \) \( 4^x \) \( 7^x \)
\( -10 \) \( 0.00097 \) \( 9.53674 \times 10^{-7} \) \( 3.54013 \times 10^{-9} \)
\( -5 \) \( 0.03125 \) \( 0.00097 \) \( 0.00006 \)
\( -1 \) \( 0.5 \) \( 0.25 \) \( 0.14285 \)
\( 0 \) \( 1 \) \(1 \) \( 1 \)
\( 1 \) \( 2 \) \( 4 \) \( 7 \)
\( 3 \) \( 8 \) \( 64 \) \( 343 \)
\( 5 \) \( 32 \) \( 1024 \) \( 16807 \)
\( 10 \) \( 1024 \) \( 1048576 \) \( 282475249 \)

graphs of exponential functions with base greater than 1



Example 2: Table of values and graphs of exponential functions with base less than 1
A table of values and the graphs of the exponential functions \( 0.2^x \), \( 0.5^x \) and \( 0.8^x\) are shown below

\( x \) \( 0.2^x \) \( 0.5^x \) \( 0.8^x \)
\( -10 \) \( 9765625 \) \( 1024 \) \( 9.31323 \)
\( -5 \) \( 3125 \) \( 32 \) \( 3.05175 \)
\( -1 \) \( 5 \) \( 2 \) \(1.25\)
\( 0 \) \( 1 \) \(1 \) \( 1 \)
\( 1 \) \( 0.2 \) \( 0.5 \) \( 0.8 \)
\( 3 \) \( 0.008 \) \( 0.125 \) \( 0.512 \)
\( 5 \) \( 0.00032 \) \( 0.03125 \) \( 0.32768 \)
\( 10 \) \( 1.024 \times 10^{-7} \) \( 0.0009765625 \) \( 0.1073741824 \)

graphs of exponential functions with base less than 1



Properties of the exponential functions

From the above values and graphs we conclude the following properties
1) The domain of any exponential function of the form \( f(x) = B^{x} \) is the set of all real numbers

2) If \( B \gt 1\), the exponential function \( f(x) = B^{x} \)
increases as x increases which may written using the limits as:    \( \lim_{x \to +\infty} f(x) = + \infty \)
and
approaches zero as x decreases which may be written using limts as:    \( \lim_{x \to -\infty} f(x) = 0 \)

3) If \( 0 \lt B \lt 1\), the exponential function \( f(x) = B^{x} \)
approaches zero as x increases which may written using the limits as:    \( \lim_{x \to +\infty} f(x) = 0 \)
and
increases as x decreases which may written using the limits as:    \( \lim_{x \to -\infty} f(x) = + \infty \)

4) Since \( f(x) = B^{x} \) approaches zero as x decreases for \( B \gt 1 \) and as x increases for \( 0 \lt B \lt 1 \), the exponential function \( f(x) = B^{x} \) has a horizontal asymptote y = 0.

5) All exponential functions of the form \( f(x) = B^{x} \) has a y intercept at (0 , 1) and they have no x intercept.

6) The range of exponential functions \( f(x) = B^{x} \) is the set of all positive real numbers which is given by the interval: \( (0 , +\infty) \)
A comprehensive set of questions on how to find the range of exponential functions is included in this website.



Change any exponential functions \( B^x \) to a natural exponential of the form \(e^{kx}\)

Let \( f(x) = B^x \) be an exponential of base B.
Use the rule of natural logarithm \( ln(e^x) = x \) to write
\( B = e^{ln(B)} \)
\( B^x = (e^{ln(B)})^x \)
Use the exponential rule \( (a^m)^n = a^{m n} \) to rewrite the above as
\( B^x = e^{ln(B) x} \)
Conclusion \[ B^x = e^{ln(B) x} \]
In calculus, it is better to change exponentials to natural exponentials because the derivative and integral of natural exponentials are studied in details.



Explore the more general exponential functions using an app

Input boxes are used to change parameters included in the definition of the more general exponential function of the form \[ f(x) = a B^{b(x+c)} + d \]
The values of the coefficients a, b, c, d, and the base B may be changed by small increments to see the effects on the properties of the graph of the function. This makes this interactive tutorial very helpful and leads to a deep understanding of the behavior of the graph of the exponential functions.


a =
b =
c =
d =
B =
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  1. Set a to 1, b to 1, c to 0, d to 0 and the base B to 2. Does the graph of function f increase or decrease?
  2. Set a to 1, b to 1, c to 0, d to 0 and the base B to 4. Does the graph of function f increase or decrease?
  3. Set a to 1, b to 1, c to 0, d to 0 and the base B to 0.4. Does the graph of function f increase or decrease?
  4. Set a to 1, b to 1, c to 0, d to 0 and the base B to 0.9. Does the graph of function f increase or decrease?

Answers to above questions.

Increase and Decrease of the Exponential Functions

Set a to 1, b to 1 , c to 0, d to 0 and change base B so that B greater than 1. Note that as long as B greater than 1, the exponential function Bx increases throughout its domain which is the set of all real numbers.
Set a to 1, b to 1 , c to 0, d to 0 and change base B so that 0 < B < 1. Note that as long as 0 < B < 1, the exponential function Bx decreases throughout its domain.



Range and Horizontal Asymptote of the Exponential Functions

Set a to 1, b to 1, c and d to zero. Set base B values greater than 1 and note the following: as x increases, Bx increases without bound (zoom in and out if necessary) and as x decreases Bx approaches zero but is never equal to zero. The graph follows the x axis. The range of Bx is given by the interval (0 , + infinity). The x axis (y = 0) is the horizontal asymptote.
Set a to 1, b to 1, c and d to zero. Set base B to values smaller than 1 and note the following: as x decreases, Bx increases without bound (zoom in and out if necessary) and as x increases Bx approaches zero but is never equal to zero. The graph follows the x axis. The range of Bx is given by the interval (0 , + infinity). The x axis (y = 0) is the horizontal asymptote.



Shifting, Scaling and Reflection of the Exponential Functions

We now investigate the effects of parameters a, b, c and d on the properties of the graph of function f defined by:

\[ f(x) = a B^{b(x+c)} + d \] Set B = e, b = 1, c = 0 and d = 0 and Explore the effects of parameter a (vertical scaling) on the graph of f.
Set a = 1, c = 0, d = 0 and B = e and Explore the effects of parameter b (horizontal scaling) on the graph of f.
Set a = 1, b = 1, d = 0 and B = e and Explore the effects of parameter c (horizontal shift) on the graph of f.
set B,a,b,c to values of your choice, change d and explain how it affects the horizontal asymptote and the range of f.
What parameter(s) affect the y intercept? Do you think the graph of this function will always have a y intercept? Explain analytically.
What parameter(s) affect the x intercept? Do you think the graph of this function will always have an x intercept? Explain analytically.

Answers to above questions.

You may want to work through a tutorial on finding Exponential Function Given its Graph. It is a tutorial that complements the one on this page.

More References and Links

Calculate Exponentials and Logarithms to any Base.
Graphing Exponential Functions
natural logarithm
natural exponential
Exponential and Logarithmic Functions.

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