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L22 Exponential Functions
Statement: If a and x are real numbers with a > 0 and
a โ 1 , then
x y = a is a uniquely defined real number.
Laws of Exponents
If s , t , a , and b are real numbers, with a > 0 , b > 0 ,
then
s t s t a a a
โ
=
s s t t
a a a
โ = ( )
s t st a = a
0 a = 1
s s s ab = a โ
b
s (^) s
s
a a
b b
s s a (^) s a a
โ โ^ โ
The exponential function with the base a is a
function of the form
( )
x f x = a ,
where a > 0 and a โ 1.
The domain of f is the set of all real numbers.
Example: For ( )
x f x = ca , ( a > 0 , a โ 1 , c โ 0 ) evaluate
f x
f x
Example: For f ( ) x = ax + b ( a โ 0 ), evaluate
f ( x + 1) โ f ( ) x =
Comparing Exponential and Linear Models:
For an exponential model , ( )
x f x = ca ( a > 0 , a โ 1 ,
c โ 0 ), for unit increases in the input, the output changes
by a factor of a (base)
f ( x + 1) = f ( ) x โ
a.
For a linear model , f ( ) x = ax + b ( a โ 0 ), for unit
increases in the input, the output changes by an additive a
(slope)
f ( x + 1) = f ( ) x + a.
(b) Size of a Population:
Under ideal conditions a certain bacteria population is
known to double every hour. Suppose that there are
initially 100 bacteria. Write the equation for the size of
the population N after t hours.
Graphing Exponential Functions
x f x =
x x f x
Note: Use these graphs as templates for graphing
( )
x f x = a with a > 1 (on the left) and 0 < a < 1
(on the right).
Properties of the Function ( )
x f x = a ( a > 0, a โ 1 ):
- Domain: Range:
- Points
a
, (0,1), and (1, a ) are on the graph.
- The line y = 0 is the horizontal asymptote:
if a > 1 , then 0
x a โ as x โ โโ
if 0 < a < 1 , then 0
x a โ as x โ +โ
x f x = a is:
increasing, if a > 1 ;
decreasing, if 0 < a < 1.
x f x = a is a one-to-one function.
Recall: f ( x ) is a one-to-one function means that
f ( u ) = f ( ) v โ u = v
Therefore, an equivalent form of the property 5 is
u v a = a โ u = v.
This property is useful for solving some exponential
equations.
Example: Solve the equations.
1 1 1
โ x x + โ โ โ โ โ โ =โ โ โ โ โ โ
1 1 9 27 (3)
x โ x
Graphing
x y = e :
x
x y = e
e 0. e
โ = โ
0 e = 1
(^1) e โ 2.
2
2 e โ7.
Example: Graph the function. Find its domain and range.
1 3
x y e
= โ +
