The Exponential âeâ and Natural Logarithmic Functions ...
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions KEY f (x) = ex 1. Complete the table and graph for
, then list the characteristics of this function.
f (x) = ex x
f (x )
Domain:
{x | x }, or (, )
Range:
{y | y > 0}, or (0, )
Asymptote :
y=0
End behavior:
As x, f(x). As x , f(x)0.
-2
e-2 =
0.13534
-1
e-1 =
0.36788
0
e0 =
1
1
e1 =
2.7183
2
e2 =
7.3891
3
e3 =
20.086
f ( x ) = ln x 2. Complete the table and graph for
, then list the characteristics of this function.
f ( x ) = ln x f (x )
x 0
[error]
0.5 1
-0.6931 0
1.5
.40547
2
.69315
3
1.0986
Domain:
{x | x > 0}, or (0, )
Range:
{y | y }, or (, )
Asymptote :
x=0
End behavior:
As x, f(x). As x0, f(x) .
3. Discuss any differences or similarities between the two functions. Sample answers: The two functions are inverses of one another. Consequently: * The graphs are reflections of one another over the line y = x. * If f(x) = ex contains the point (a,b), then f(x) = ln x contains the point (b,a). * While f(x) = ex is only defined for y > 0, f(x) = ln x is only defined for x > 0. ©2012, TESCCC
11/05/12
page 1 of 8
Precalculus HS Mathematics Unit: 09 Lesson: 01
©2012, TESCCC
11/05/12
page 2 of 8
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions KEY 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=
f (x ) If and f (g ( x )) =
g (f ( x ))
, or
y=
e^(ln(x))
g (x )
y=
are inverses, then x
, or
y=
f (x )
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 )) =
ln(e^(x))
g (x ) are inverses, then x
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 3 of 8
Precalculus HS Mathematics Unit: 09 Lesson: 01
Restriction: x > 0 Or, e^(ln(x)) = x only when x > 0. This is because x > 0 is the domain restriction of the natural log function, which is in the “inside” of the exponential e function in this example.
©2012, TESCCC
Restriction: none Or, ln(e x) = x for any value of x. This is because the domain of the exponential “e” function is all real numbers, and this is in the “inside” of the natural log function in this example.
11/05/12
page 4 of 8
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions KEY 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)
8)
10)
12)
©2012, TESCCC
Column A f ( x ) = ( e x )2
7)
Column B f ( x ) = 7.389e x
x f(x) 0 1
x 0
f(x) 7.389
1 7.389
1
20.09
2 54.598
2
54.60
9)
f (x ) = ex
f ( x ) = 0.1353e x
x 0
f(x) 1
x 0
f(x) 0.135
1
1.649
1
0.368
2
2.718
2
1
f (x) =
11)
2 ex
x 0
f(x) 2
1
0.736
2
0.271 13)
f (x ) = ex +2
f ( x ) = e0.5 x x 0
f(x) 1
1
1.649
2
2.718
f ( x ) = e2x
x 0
f(x) 7.389
x 0
1
1
20.09
1
7.389
2
54.60
2
54.598
11/05/12
f(x)
page 5 of 8
Precalculus HS Mathematics Unit: 09 Lesson: 01
14)
15)
f ( x ) = ex −2
f ( x ) = 2e − x
x 0
f(x) 0.135
x 0
f(x) 2
1
0.368
1
0.736
2
1
2
0.271
Note: Pairs of functions are equivalent (6 & 13, 7 & 12, 8 & 11, 9 & 14, 10 & 15).
©2012, TESCCC
11/05/12
page 6 of 8
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions KEY 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)
18)
20)
22)
©2012, TESCCC
Column P f ( x ) = ln( 2 x )
17)
Column Q f ( x ) = 21 ln x
x 1
f(x) 0.693
x 1
0
2
1.386
2
0.347
3
1.792
3
0.549
f ( x ) = ln( 2x )
19)
f(x)
f ( x ) = 2 ln x
x 1
f(x) -.693
x 1
0
2
0
2
1.386
3
0.405
3
2.197
f ( x ) = ln( x1 )
21)
f(x)
f ( x ) = ln x + 0.693
x 1
f(x) 0
x 1
f(x) 0.693
2
-.693
2
1.386
3
-1.10
3
1.792
23)
f ( x ) = ln( x 2 )
f ( x ) = ln x − 0.693
x 1
f(x) 0
x 1
f(x) -.693
2
1.386
2
0
3
2.197
3
0.405
11/05/12
page 7 of 8
24)
( )
25)
f ( x ) = ln x
f ( x ) = − ln x
0
x 1
0
2
0.347
2
-.693
3
0.549
3
-1.10
x 1
f(x)
Precalculus HS Mathematics Unit: 09 Lesson: 01
f(x)
Note: Pairs of functions are equivalent (16 & 21, 17 & 24, 18 & 23, 19 & 22, 20 & 25).
©2012, TESCCC
11/05/12
page 8 of 8
The Exponential “e” and Natural Logarithmic Functions KEY f (x) = ex 1. Complete the table and graph for
, then list the characteristics of this function.
f (x) = ex x
f (x )
Domain:
{x | x }, or (, )
Range:
{y | y > 0}, or (0, )
Asymptote :
y=0
End behavior:
As x, f(x). As x , f(x)0.
-2
e-2 =
0.13534
-1
e-1 =
0.36788
0
e0 =
1
1
e1 =
2.7183
2
e2 =
7.3891
3
e3 =
20.086
f ( x ) = ln x 2. Complete the table and graph for
, then list the characteristics of this function.
f ( x ) = ln x f (x )
x 0
[error]
0.5 1
-0.6931 0
1.5
.40547
2
.69315
3
1.0986
Domain:
{x | x > 0}, or (0, )
Range:
{y | y }, or (, )
Asymptote :
x=0
End behavior:
As x, f(x). As x0, f(x) .
3. Discuss any differences or similarities between the two functions. Sample answers: The two functions are inverses of one another. Consequently: * The graphs are reflections of one another over the line y = x. * If f(x) = ex contains the point (a,b), then f(x) = ln x contains the point (b,a). * While f(x) = ex is only defined for y > 0, f(x) = ln x is only defined for x > 0. ©2012, TESCCC
11/05/12
page 1 of 8
Precalculus HS Mathematics Unit: 09 Lesson: 01
©2012, TESCCC
11/05/12
page 2 of 8
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions KEY 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=
f (x ) If and f (g ( x )) =
g (f ( x ))
, or
y=
e^(ln(x))
g (x )
y=
are inverses, then x
, or
y=
f (x )
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 )) =
ln(e^(x))
g (x ) are inverses, then x
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 3 of 8
Precalculus HS Mathematics Unit: 09 Lesson: 01
Restriction: x > 0 Or, e^(ln(x)) = x only when x > 0. This is because x > 0 is the domain restriction of the natural log function, which is in the “inside” of the exponential e function in this example.
©2012, TESCCC
Restriction: none Or, ln(e x) = x for any value of x. This is because the domain of the exponential “e” function is all real numbers, and this is in the “inside” of the natural log function in this example.
11/05/12
page 4 of 8
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions KEY 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)
8)
10)
12)
©2012, TESCCC
Column A f ( x ) = ( e x )2
7)
Column B f ( x ) = 7.389e x
x f(x) 0 1
x 0
f(x) 7.389
1 7.389
1
20.09
2 54.598
2
54.60
9)
f (x ) = ex
f ( x ) = 0.1353e x
x 0
f(x) 1
x 0
f(x) 0.135
1
1.649
1
0.368
2
2.718
2
1
f (x) =
11)
2 ex
x 0
f(x) 2
1
0.736
2
0.271 13)
f (x ) = ex +2
f ( x ) = e0.5 x x 0
f(x) 1
1
1.649
2
2.718
f ( x ) = e2x
x 0
f(x) 7.389
x 0
1
1
20.09
1
7.389
2
54.60
2
54.598
11/05/12
f(x)
page 5 of 8
Precalculus HS Mathematics Unit: 09 Lesson: 01
14)
15)
f ( x ) = ex −2
f ( x ) = 2e − x
x 0
f(x) 0.135
x 0
f(x) 2
1
0.368
1
0.736
2
1
2
0.271
Note: Pairs of functions are equivalent (6 & 13, 7 & 12, 8 & 11, 9 & 14, 10 & 15).
©2012, TESCCC
11/05/12
page 6 of 8
Precalculus HS Mathematics Unit: 09 Lesson: 01
The Exponential “e” and Natural Logarithmic Functions KEY 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)
18)
20)
22)
©2012, TESCCC
Column P f ( x ) = ln( 2 x )
17)
Column Q f ( x ) = 21 ln x
x 1
f(x) 0.693
x 1
0
2
1.386
2
0.347
3
1.792
3
0.549
f ( x ) = ln( 2x )
19)
f(x)
f ( x ) = 2 ln x
x 1
f(x) -.693
x 1
0
2
0
2
1.386
3
0.405
3
2.197
f ( x ) = ln( x1 )
21)
f(x)
f ( x ) = ln x + 0.693
x 1
f(x) 0
x 1
f(x) 0.693
2
-.693
2
1.386
3
-1.10
3
1.792
23)
f ( x ) = ln( x 2 )
f ( x ) = ln x − 0.693
x 1
f(x) 0
x 1
f(x) -.693
2
1.386
2
0
3
2.197
3
0.405
11/05/12
page 7 of 8
24)
( )
25)
f ( x ) = ln x
f ( x ) = − ln x
0
x 1
0
2
0.347
2
-.693
3
0.549
3
-1.10
x 1
f(x)
Precalculus HS Mathematics Unit: 09 Lesson: 01
f(x)
Note: Pairs of functions are equivalent (16 & 21, 17 & 24, 18 & 23, 19 & 22, 20 & 25).
©2012, TESCCC
11/05/12
page 8 of 8
