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Exponential Function - Definition
• The exponential function
is given by ...
f (x)  b
where x is any real number
and ...
b  0, b  1
• Example 1:
x
f (x)  3
x
f is an exponential function where the base is 3 and the
exponent is x.
• Example 2:
g( x )  (3) x
g is not an exponential function since b = -3 < 0.
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Exponential Function - Definition
• Why is it necessary to have the restrictions on the
base b? Consider the following examples.
• Example 3:
f (x)  1
x
For all values of x, f(x) = 1. This reduces the equation to
the simple linear equation, y = 1, which is not exponential.
• Example 4:
g( x )  (3)
x
Using the table feature on a
graphing calculator yields ...
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Slide 2
Exponential Function - Definition
• Note that the data represents points on a graph that
keep jumping from positive to negative to positive
y-values. The points on the graph are not “connected”
like most graphs are.
• Another problem is what
to do when x = 1/2, where
1
g   (3)   3
2
1
2
which is complex.
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Slide 3
Exponential Function - Definition
• Remember that the exponential function is quite
different from a polynomial function.
Polynomial Function
Exponential Function
f (x)  x  3x  2
f (x)  4  7
x
2
• The polynomial function
has a variable base ...
and a constant exponent.
• The exponential function
has a constant base ...
and a variable exponent.
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Slide 4
Exponential Function - Definition
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