The Exponential âeâ and Natural Logarithmic Functions
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions f (x) = ex 1. Complete the table and graph for
, then list the characteristics of this function.
f (x) = ex Domain:
f (x )
x -2
e-2 =
-1
e-1 =
0
e0 =
1
e1 =
2
e2 =
3
e3 =
Range: Asymptote : End behavior:
f ( x ) = ln x 2. Complete the table and graph for
, then list the characteristics of this function.
f ( x ) = ln x Domain:
f (x )
x 0
Range:
0.5 Asymptote:
1 1.5
End behavior:
2 3
3. Discuss any differences or similarities between the two functions.
©2012, TESCCC
11/05/12
page 1 of 7
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions y = ex You may have noticed (or remembered) that the functions
y = ln x and
are inverses of one y = ex another. While being inverses gives these functions certain graphical properties, the fact that y = ln x and are inverses also creates some important algebraic implications. Graph the composites of the inverse functions. Answer the questions that follow, concerning the composites of the inverse functions. 4.
f (x) = ex If
5.
,
f (x) = ex If
,
g ( x ) = ln x and
g ( x ) = ln x ,
and
then sketch the graph of
then sketch the graph of
f (g ( x )) y=
g (f ( x )) , or
y=
f (x ) If and f (g ( x )) =
y=
, or
y=
g (x )
f (x ) are inverses, then
According to the graph, what restrictions (if any) must be used in order to say that the following is true? Explain. eln x = x
©2012, TESCCC
,
If and g (f ( x )) =
g (x ) are inverses, then
According to the graph, what restrictions (if any) must be used in order to say that the following is true? Explain. ln e x = x
( )
11/05/12
page 2 of 7
Precalculus HS Mathematics Unit: 09 Lesson: 01
©2012, TESCCC
11/05/12
page 3 of 7
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions f (x) = ex The following functions are transformations of the exponential parent function . As your teacher instructs, complete the graphs and tables in either Column A or Column B (but not both). 6)
Column A f ( x ) = (e x )2 x 0
8)
f(x)
x 0
1
1
2
2 9)
f (x) = ex x 0
10)
7)
Column B f ( x ) = 7.389e x
f ( x ) = 0.1353e x x 0
f(x)
1
1
2
2
f (x) = x 0
11)
2 ex
f(x)
f ( x ) = e 0.5 x x 0
f(x)
f(x)
f(x)
1
1
2
2 12)
x 0
©2012, TESCCC
13)
f ( x ) = e x +2 f(x)
f ( x ) = e2 x x 0
1
1
2
2
11/05/12
f(x)
page 4 of 7
Precalculus HS Mathematics Unit: 09 Lesson: 01
14)
x 0
©2012, TESCCC
15)
f ( x ) = e x −2 f(x)
f ( x ) = 2e − x x 0
1
1
2
2
11/05/12
f(x)
page 5 of 7
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions f ( x ) = ln x The following functions are transformations of the exponential parent function . As your teacher instructs, complete the graphs and tables in either Column P or Column Q (but not both). 16)
Column P f ( x ) = ln( 2 x ) x 1
18)
3
3
f ( x ) = ln ( 2x )
x 1
f(x)
3
3
f ( x ) = ln ( 1x )
21)
f(x)
f ( x ) = ln x + 0.693
x 1
f(x)
f(x)
f ( x ) = 2 ln x
2
2
2
3
3
f(x)
23) f ( x ) = ln x − 0.693
f ( x ) = ln( x 2 )
x 1
f(x)
2
2
3
3
( )
25)
f ( x ) = ln x x 1
©2012, TESCCC
19)
2
x 1
24)
x 1 2
x 1
22)
f(x)
Column Q f ( x ) = 21 ln x
2
x 1
20)
17)
f ( x ) = − ln x x 1
f(x)
2
2
3
3 11/05/12
f(x)
f(x)
page 6 of 7
Precalculus HS Mathematics Unit: 09 Lesson: 01
©2012, TESCCC
11/05/12
page 7 of 7
The Exponential “e” and Natural Logarithmic Functions f (x) = ex 1. Complete the table and graph for
, then list the characteristics of this function.
f (x) = ex Domain:
f (x )
x -2
e-2 =
-1
e-1 =
0
e0 =
1
e1 =
2
e2 =
3
e3 =
Range: Asymptote : End behavior:
f ( x ) = ln x 2. Complete the table and graph for
, then list the characteristics of this function.
f ( x ) = ln x Domain:
f (x )
x 0
Range:
0.5 Asymptote:
1 1.5
End behavior:
2 3
3. Discuss any differences or similarities between the two functions.
©2012, TESCCC
11/05/12
page 1 of 7
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions y = ex You may have noticed (or remembered) that the functions
y = ln x and
are inverses of one y = ex another. While being inverses gives these functions certain graphical properties, the fact that y = ln x and are inverses also creates some important algebraic implications. Graph the composites of the inverse functions. Answer the questions that follow, concerning the composites of the inverse functions. 4.
f (x) = ex If
5.
,
f (x) = ex If
,
g ( x ) = ln x and
g ( x ) = ln x ,
and
then sketch the graph of
then sketch the graph of
f (g ( x )) y=
g (f ( x )) , or
y=
f (x ) If and f (g ( x )) =
y=
, or
y=
g (x )
f (x ) are inverses, then
According to the graph, what restrictions (if any) must be used in order to say that the following is true? Explain. eln x = x
©2012, TESCCC
,
If and g (f ( x )) =
g (x ) are inverses, then
According to the graph, what restrictions (if any) must be used in order to say that the following is true? Explain. ln e x = x
( )
11/05/12
page 2 of 7
Precalculus HS Mathematics Unit: 09 Lesson: 01
©2012, TESCCC
11/05/12
page 3 of 7
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions f (x) = ex The following functions are transformations of the exponential parent function . As your teacher instructs, complete the graphs and tables in either Column A or Column B (but not both). 6)
Column A f ( x ) = (e x )2 x 0
8)
f(x)
x 0
1
1
2
2 9)
f (x) = ex x 0
10)
7)
Column B f ( x ) = 7.389e x
f ( x ) = 0.1353e x x 0
f(x)
1
1
2
2
f (x) = x 0
11)
2 ex
f(x)
f ( x ) = e 0.5 x x 0
f(x)
f(x)
f(x)
1
1
2
2 12)
x 0
©2012, TESCCC
13)
f ( x ) = e x +2 f(x)
f ( x ) = e2 x x 0
1
1
2
2
11/05/12
f(x)
page 4 of 7
Precalculus HS Mathematics Unit: 09 Lesson: 01
14)
x 0
©2012, TESCCC
15)
f ( x ) = e x −2 f(x)
f ( x ) = 2e − x x 0
1
1
2
2
11/05/12
f(x)
page 5 of 7
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions f ( x ) = ln x The following functions are transformations of the exponential parent function . As your teacher instructs, complete the graphs and tables in either Column P or Column Q (but not both). 16)
Column P f ( x ) = ln( 2 x ) x 1
18)
3
3
f ( x ) = ln ( 2x )
x 1
f(x)
3
3
f ( x ) = ln ( 1x )
21)
f(x)
f ( x ) = ln x + 0.693
x 1
f(x)
f(x)
f ( x ) = 2 ln x
2
2
2
3
3
f(x)
23) f ( x ) = ln x − 0.693
f ( x ) = ln( x 2 )
x 1
f(x)
2
2
3
3
( )
25)
f ( x ) = ln x x 1
©2012, TESCCC
19)
2
x 1
24)
x 1 2
x 1
22)
f(x)
Column Q f ( x ) = 21 ln x
2
x 1
20)
17)
f ( x ) = − ln x x 1
f(x)
2
2
3
3 11/05/12
f(x)
f(x)
page 6 of 7
Precalculus HS Mathematics Unit: 09 Lesson: 01
©2012, TESCCC
11/05/12
page 7 of 7
