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5.1
EXPONENTIAL FUNCTIONS
So far, we have studied polynomial functions and rational functions. These
are examples
of algebraic functions. We now turn our attention to exponential and
logarithmic
functions, which are called transcendental functions. A transcendental
function is
a function that does not satisfy a polynomial equation whose coefficients are
themselves
roots of polynomials, in contrast to an algebraic function, which does
satisfy such an
equation. In other words, a transcendental function is a function that
"transcends" algebra in the sense that it cannot be expressed in terms of a
finite sequence
of the algebraic operations of addition, multiplication, and root extraction.
An exponential function with base a is denoted as f x   a x where a  0 , a
 1, and x
is any real number.
Example 1 – Evaluating Exponential Expressions
Use a calculator to evaluate each expression.
a) 2 3.1
b) 2
c) 125 / 7
3/ 2
d) 0.6
Graphing Exponential Functions
To graph exponential functions, choose some values of x, and find the
corresponding
values of f ( x) . Plot the points and draw a smooth curve through them.
Example 2 – Graph each function:
a)
b)
c)
d)
f x   3 x
f x   3 x
f  x   3 x 2
f x   3 x  1
a)
x
-3
-2
-1
0
1
2
3
f x   3 x
1/27
1/9
1/3
1
3
9
27
b)
x
-3
-2
-1
0
1
2
3
c)
x
0
1
2
3
4
d)
x
-2
-1
0
1
2
f x   3 x
27
9
3
1
1/3
1/9
1/27
f  x   3 x 2
1/9
1/3
1
3
9
f x   3 x  1
10/9
4/3
2
4
10
Solving Exponential Equations
To solve an exponential equation, use the steps below:
Step 1 – Each side must have the same base. If the two sides do not have the
same
base, express each as a power of the same base, if possible.
Step 2 – Simplify exponents if necessary, using rules of exponents.
Step 3 – Set exponents equal.
Step 4 – Solve the equation obtained in Step 3.
Example 3 – Solve each equation.
x
9  27
x
1
   64
4
4 x 2 
1
64
 3
x 9
 27 x
The Natural Base e
An irrational number, symbolized by the letter e, appears as the base in many
applied
exponential functions. This irrational number is approximately equal to 2.72.
More
accurately, e  2.71828 . The number e is called the natural base. The
function
f  x   e x is called the natural exponential function.
m
1

The number e can be approximated as e  1   for large values of m .
 m
Using a Graphing or Scientific Calculator to Evaluate Powers of e
Example 4 – use a calculator to evaluate each:
a)
b)
c)
d)
e2
e 0.2
e 0.25
e
The number e lies between 2 and 3. Therefore, the graph of f  x   e x lies
between the
graphs of f  x   2x and f  x   3x , as seen below.
2.5
2
1.5
1
0.5
-2
-1
1
2
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MATH 1314 5.1 exponential functions.doc
Catalog Description: Course Outline for Mathematics 20 PRE-CALCULUS
MATHEMATICS •
Note. - Faculty
Pre-Calculus Mathematics
MATH 2413 - Calculus I
Algebra 2 – Things You Will Definitely Need to Know For the
Pre-Calculus Mathematics
learner outcomes
Math 103 - Cuyamaca College
Numbers - Keith Enevoldsen`s Think Zone
Transcendental
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