Exponential Functions and their Applications
The functions that involve some combination of basic arithmetic operations ,
powers,
or roots are called algebraic functions . Most of the functions we studied
so far are
algebraic functions. In this chapter we turn to exponential and logarithmic
functions .
These functions are used to describe phenomena ranging from growth of
investments
to the decay of radioactive materials, which cannot be described with algebraic
functions. Since the exponential and logarithmic functions transcend what can be
described with algebraic functions, they are called transcendental
functions.
The Definition
In algebraic functions such as
, and 
the base is a variable and the exponent is constant. For the exponential
functions
the base is constant the exponent is variable. The fol lowing functions are
exponential
functions:
, and 
Definition: Exponential Function
An exponential function with base a is a function of the form
where a and x are real numbers such that a > 0 and a ≠ 1.
We rule out a = 1 in the definition because 
is the constant function
f(x) = 1. Negative numbers are not used as bases because powers of such are not
real numbers. Functions that have variable exponents but are not exactly in the
form
of the definition may also be called exponential functions.
Examples
Let 
. Find the following values . Check your
answers with a calculator .
a. f(0) =
b. f(1) =
c. f(4) =
d. f(-1) =
e. f(1/2) =
f. f(3/5) =
Laws of Exponents
Examples – Write each function in the form
or 
, for a
suitable constant k.
Solving Exponential Equations
For a > 0 and a ≠ 1, If 
, then x = y.
• When each side of an exponential equation can be written as a power of the
same base, set the exponents equal to each other and solve the resulting
equation.
• The one-to-one property is used in solving simple exponential equations. For
example, to solve 
, we recall that
. So, the equation becomes
.
By the one-to-one property, x = 3 is the only solution .
Examples – Solving the following equations for x.
More Examples – Finding missing factorings. The
following may be factored.
So what do these graphs look like and what properties
do they exhibit?
Domain of an Exponential Function
The domain of for
a > 0 and a ≠ 1 is the set of all real numbers.
Graphing Exponential Functions
Even though the domain of an exponential function is the set of real numbers,
for
ease of computation, we generally choose only rational numbers for x to find
ordered pairs on the graph of the function.
Examples
Sketch the graph of each function by finding at least three ordered pairs on the
graph. State the domain, range, and whether the function is increasing or
decreasing.
1. 
Find some ordered pairs satisfying as
follows:
Ordered pairs are:
Domain is:
Range is:
Is this function increasing or decreasing?
Section 4.2 – The Exponential Function
What is ?
Examples – Write each function the form 
for a suitable constant k.
Examples – Solving the following equations for x.
Homework
Section 4.1: Page 241 #1, 5, 9, 13, 17 – 33 odds.
Section 4.2: Page 246 #9 – 17 odds.
