# 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

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

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

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

## 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 = 1 b = 1 c = 0 d = 0 B = 2
>

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?

## 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.

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.

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