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73 Power Functions and Function Operations
Goals
1) Perform operations with functions including power functions
2) Use power functions and function operations to solver real-life problems
Evaluate each function for when x = -2
bull f(x) = 2x + 5
bull f(x) = 2(-2) + 5bull f(x) = -4 + 5bull f(x) = 1
bull g(x) = x2 ndash 1
bull g(x) = (-2)2 ndash 1bull g(x) = 4 -1bull g(x) = 3
3
Domainbull The numbers that CAN be plugged in
for x in a functionndash Have to look at beginning function through the ending function for domain
bull Try to find the values that CANNOT be plugged in to determine domain1 Canrsquot divide by zero (undefined)
2 Canrsquot take even root of a negative number
bull If there are no restrictions on x then the domain is ldquoall realsrdquo
Operations with Functionsbull The four basic operations can be applied
to any function
bull Letrsquos perform the basic operations with this example f(x) = 2x + 5 g(x) = x2 - 1
Addition of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
Is f(x) + g(x) = g(x) + f(x)
Subtraction of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) ndash g(x) = (2x + 5) ndash (x2 - 1)
= 2x + 5 ndash x2 + 1
= -x2 + 2x + 6
Domain = ldquoall realsrdquo
Is f(x) ndash g(x) = g(x) ndash f(x)
Multiplication of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5)(x2 - 1)
= 2x3 - 2x + 5x2 - 5
= 2x3 +5x2 - 2x - 5
Domain = ldquoall realsrdquo
Is f(x) g(x) = g(x) f(x)
Division of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5) (x2 - 1)
Domain =
Is f(x) g(x) = g(x) f(x)
2
2 5
1
x
x
1 1x or
9
Composition
bull Plug one function into another
ndashPut one function in place of x
( ) ( ( ))h x f g x
Composition of Functions Example 1
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(g(x))
f(x2 + 5) = 2(x2 + 5) + 3
= 2x2 + 10 + 3
= 2x2 + 13
Domain = ldquoall realsrdquo
Composition of Functions Example 2
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find g(f(x))
g(2x + 3) = (2x + 3)2 + 5
= (2x + 3)(2x + 3) + 5
= 4x2 + 6x + 6x + 9 + 5
= 4x2 + 12x +14
Domain = ldquoall realsrdquo
Composition of Functions Example 3
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(f(x))
f(2x + 3) = 2(2x + 3) + 3
= 4x + 6 + 3
= 4x + 9
Domain = ldquoall realsrdquo
One More Example
bull f(x) = 3x and g(x) = x14
bull f(x) g(x)
Domain = ldquopositive real srdquo
1
4
3x
x
3
43x
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
Evaluate each function for when x = -2
bull f(x) = 2x + 5
bull f(x) = 2(-2) + 5bull f(x) = -4 + 5bull f(x) = 1
bull g(x) = x2 ndash 1
bull g(x) = (-2)2 ndash 1bull g(x) = 4 -1bull g(x) = 3
3
Domainbull The numbers that CAN be plugged in
for x in a functionndash Have to look at beginning function through the ending function for domain
bull Try to find the values that CANNOT be plugged in to determine domain1 Canrsquot divide by zero (undefined)
2 Canrsquot take even root of a negative number
bull If there are no restrictions on x then the domain is ldquoall realsrdquo
Operations with Functionsbull The four basic operations can be applied
to any function
bull Letrsquos perform the basic operations with this example f(x) = 2x + 5 g(x) = x2 - 1
Addition of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
Is f(x) + g(x) = g(x) + f(x)
Subtraction of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) ndash g(x) = (2x + 5) ndash (x2 - 1)
= 2x + 5 ndash x2 + 1
= -x2 + 2x + 6
Domain = ldquoall realsrdquo
Is f(x) ndash g(x) = g(x) ndash f(x)
Multiplication of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5)(x2 - 1)
= 2x3 - 2x + 5x2 - 5
= 2x3 +5x2 - 2x - 5
Domain = ldquoall realsrdquo
Is f(x) g(x) = g(x) f(x)
Division of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5) (x2 - 1)
Domain =
Is f(x) g(x) = g(x) f(x)
2
2 5
1
x
x
1 1x or
9
Composition
bull Plug one function into another
ndashPut one function in place of x
( ) ( ( ))h x f g x
Composition of Functions Example 1
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(g(x))
f(x2 + 5) = 2(x2 + 5) + 3
= 2x2 + 10 + 3
= 2x2 + 13
Domain = ldquoall realsrdquo
Composition of Functions Example 2
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find g(f(x))
g(2x + 3) = (2x + 3)2 + 5
= (2x + 3)(2x + 3) + 5
= 4x2 + 6x + 6x + 9 + 5
= 4x2 + 12x +14
Domain = ldquoall realsrdquo
Composition of Functions Example 3
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(f(x))
f(2x + 3) = 2(2x + 3) + 3
= 4x + 6 + 3
= 4x + 9
Domain = ldquoall realsrdquo
One More Example
bull f(x) = 3x and g(x) = x14
bull f(x) g(x)
Domain = ldquopositive real srdquo
1
4
3x
x
3
43x
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
3
Domainbull The numbers that CAN be plugged in
for x in a functionndash Have to look at beginning function through the ending function for domain
bull Try to find the values that CANNOT be plugged in to determine domain1 Canrsquot divide by zero (undefined)
2 Canrsquot take even root of a negative number
bull If there are no restrictions on x then the domain is ldquoall realsrdquo
Operations with Functionsbull The four basic operations can be applied
to any function
bull Letrsquos perform the basic operations with this example f(x) = 2x + 5 g(x) = x2 - 1
Addition of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
Is f(x) + g(x) = g(x) + f(x)
Subtraction of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) ndash g(x) = (2x + 5) ndash (x2 - 1)
= 2x + 5 ndash x2 + 1
= -x2 + 2x + 6
Domain = ldquoall realsrdquo
Is f(x) ndash g(x) = g(x) ndash f(x)
Multiplication of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5)(x2 - 1)
= 2x3 - 2x + 5x2 - 5
= 2x3 +5x2 - 2x - 5
Domain = ldquoall realsrdquo
Is f(x) g(x) = g(x) f(x)
Division of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5) (x2 - 1)
Domain =
Is f(x) g(x) = g(x) f(x)
2
2 5
1
x
x
1 1x or
9
Composition
bull Plug one function into another
ndashPut one function in place of x
( ) ( ( ))h x f g x
Composition of Functions Example 1
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(g(x))
f(x2 + 5) = 2(x2 + 5) + 3
= 2x2 + 10 + 3
= 2x2 + 13
Domain = ldquoall realsrdquo
Composition of Functions Example 2
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find g(f(x))
g(2x + 3) = (2x + 3)2 + 5
= (2x + 3)(2x + 3) + 5
= 4x2 + 6x + 6x + 9 + 5
= 4x2 + 12x +14
Domain = ldquoall realsrdquo
Composition of Functions Example 3
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(f(x))
f(2x + 3) = 2(2x + 3) + 3
= 4x + 6 + 3
= 4x + 9
Domain = ldquoall realsrdquo
One More Example
bull f(x) = 3x and g(x) = x14
bull f(x) g(x)
Domain = ldquopositive real srdquo
1
4
3x
x
3
43x
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
Operations with Functionsbull The four basic operations can be applied
to any function
bull Letrsquos perform the basic operations with this example f(x) = 2x + 5 g(x) = x2 - 1
Addition of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
Is f(x) + g(x) = g(x) + f(x)
Subtraction of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) ndash g(x) = (2x + 5) ndash (x2 - 1)
= 2x + 5 ndash x2 + 1
= -x2 + 2x + 6
Domain = ldquoall realsrdquo
Is f(x) ndash g(x) = g(x) ndash f(x)
Multiplication of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5)(x2 - 1)
= 2x3 - 2x + 5x2 - 5
= 2x3 +5x2 - 2x - 5
Domain = ldquoall realsrdquo
Is f(x) g(x) = g(x) f(x)
Division of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5) (x2 - 1)
Domain =
Is f(x) g(x) = g(x) f(x)
2
2 5
1
x
x
1 1x or
9
Composition
bull Plug one function into another
ndashPut one function in place of x
( ) ( ( ))h x f g x
Composition of Functions Example 1
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(g(x))
f(x2 + 5) = 2(x2 + 5) + 3
= 2x2 + 10 + 3
= 2x2 + 13
Domain = ldquoall realsrdquo
Composition of Functions Example 2
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find g(f(x))
g(2x + 3) = (2x + 3)2 + 5
= (2x + 3)(2x + 3) + 5
= 4x2 + 6x + 6x + 9 + 5
= 4x2 + 12x +14
Domain = ldquoall realsrdquo
Composition of Functions Example 3
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(f(x))
f(2x + 3) = 2(2x + 3) + 3
= 4x + 6 + 3
= 4x + 9
Domain = ldquoall realsrdquo
One More Example
bull f(x) = 3x and g(x) = x14
bull f(x) g(x)
Domain = ldquopositive real srdquo
1
4
3x
x
3
43x
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
Addition of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
Is f(x) + g(x) = g(x) + f(x)
Subtraction of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) ndash g(x) = (2x + 5) ndash (x2 - 1)
= 2x + 5 ndash x2 + 1
= -x2 + 2x + 6
Domain = ldquoall realsrdquo
Is f(x) ndash g(x) = g(x) ndash f(x)
Multiplication of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5)(x2 - 1)
= 2x3 - 2x + 5x2 - 5
= 2x3 +5x2 - 2x - 5
Domain = ldquoall realsrdquo
Is f(x) g(x) = g(x) f(x)
Division of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5) (x2 - 1)
Domain =
Is f(x) g(x) = g(x) f(x)
2
2 5
1
x
x
1 1x or
9
Composition
bull Plug one function into another
ndashPut one function in place of x
( ) ( ( ))h x f g x
Composition of Functions Example 1
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(g(x))
f(x2 + 5) = 2(x2 + 5) + 3
= 2x2 + 10 + 3
= 2x2 + 13
Domain = ldquoall realsrdquo
Composition of Functions Example 2
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find g(f(x))
g(2x + 3) = (2x + 3)2 + 5
= (2x + 3)(2x + 3) + 5
= 4x2 + 6x + 6x + 9 + 5
= 4x2 + 12x +14
Domain = ldquoall realsrdquo
Composition of Functions Example 3
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(f(x))
f(2x + 3) = 2(2x + 3) + 3
= 4x + 6 + 3
= 4x + 9
Domain = ldquoall realsrdquo
One More Example
bull f(x) = 3x and g(x) = x14
bull f(x) g(x)
Domain = ldquopositive real srdquo
1
4
3x
x
3
43x
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
Subtraction of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) ndash g(x) = (2x + 5) ndash (x2 - 1)
= 2x + 5 ndash x2 + 1
= -x2 + 2x + 6
Domain = ldquoall realsrdquo
Is f(x) ndash g(x) = g(x) ndash f(x)
Multiplication of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5)(x2 - 1)
= 2x3 - 2x + 5x2 - 5
= 2x3 +5x2 - 2x - 5
Domain = ldquoall realsrdquo
Is f(x) g(x) = g(x) f(x)
Division of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5) (x2 - 1)
Domain =
Is f(x) g(x) = g(x) f(x)
2
2 5
1
x
x
1 1x or
9
Composition
bull Plug one function into another
ndashPut one function in place of x
( ) ( ( ))h x f g x
Composition of Functions Example 1
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(g(x))
f(x2 + 5) = 2(x2 + 5) + 3
= 2x2 + 10 + 3
= 2x2 + 13
Domain = ldquoall realsrdquo
Composition of Functions Example 2
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find g(f(x))
g(2x + 3) = (2x + 3)2 + 5
= (2x + 3)(2x + 3) + 5
= 4x2 + 6x + 6x + 9 + 5
= 4x2 + 12x +14
Domain = ldquoall realsrdquo
Composition of Functions Example 3
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(f(x))
f(2x + 3) = 2(2x + 3) + 3
= 4x + 6 + 3
= 4x + 9
Domain = ldquoall realsrdquo
One More Example
bull f(x) = 3x and g(x) = x14
bull f(x) g(x)
Domain = ldquopositive real srdquo
1
4
3x
x
3
43x
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
Multiplication of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5)(x2 - 1)
= 2x3 - 2x + 5x2 - 5
= 2x3 +5x2 - 2x - 5
Domain = ldquoall realsrdquo
Is f(x) g(x) = g(x) f(x)
Division of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5) (x2 - 1)
Domain =
Is f(x) g(x) = g(x) f(x)
2
2 5
1
x
x
1 1x or
9
Composition
bull Plug one function into another
ndashPut one function in place of x
( ) ( ( ))h x f g x
Composition of Functions Example 1
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(g(x))
f(x2 + 5) = 2(x2 + 5) + 3
= 2x2 + 10 + 3
= 2x2 + 13
Domain = ldquoall realsrdquo
Composition of Functions Example 2
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find g(f(x))
g(2x + 3) = (2x + 3)2 + 5
= (2x + 3)(2x + 3) + 5
= 4x2 + 6x + 6x + 9 + 5
= 4x2 + 12x +14
Domain = ldquoall realsrdquo
Composition of Functions Example 3
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(f(x))
f(2x + 3) = 2(2x + 3) + 3
= 4x + 6 + 3
= 4x + 9
Domain = ldquoall realsrdquo
One More Example
bull f(x) = 3x and g(x) = x14
bull f(x) g(x)
Domain = ldquopositive real srdquo
1
4
3x
x
3
43x
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
Division of Functions
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) g(x) = (2x + 5) (x2 - 1)
Domain =
Is f(x) g(x) = g(x) f(x)
2
2 5
1
x
x
1 1x or
9
Composition
bull Plug one function into another
ndashPut one function in place of x
( ) ( ( ))h x f g x
Composition of Functions Example 1
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(g(x))
f(x2 + 5) = 2(x2 + 5) + 3
= 2x2 + 10 + 3
= 2x2 + 13
Domain = ldquoall realsrdquo
Composition of Functions Example 2
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find g(f(x))
g(2x + 3) = (2x + 3)2 + 5
= (2x + 3)(2x + 3) + 5
= 4x2 + 6x + 6x + 9 + 5
= 4x2 + 12x +14
Domain = ldquoall realsrdquo
Composition of Functions Example 3
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(f(x))
f(2x + 3) = 2(2x + 3) + 3
= 4x + 6 + 3
= 4x + 9
Domain = ldquoall realsrdquo
One More Example
bull f(x) = 3x and g(x) = x14
bull f(x) g(x)
Domain = ldquopositive real srdquo
1
4
3x
x
3
43x
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
9
Composition
bull Plug one function into another
ndashPut one function in place of x
( ) ( ( ))h x f g x
Composition of Functions Example 1
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(g(x))
f(x2 + 5) = 2(x2 + 5) + 3
= 2x2 + 10 + 3
= 2x2 + 13
Domain = ldquoall realsrdquo
Composition of Functions Example 2
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find g(f(x))
g(2x + 3) = (2x + 3)2 + 5
= (2x + 3)(2x + 3) + 5
= 4x2 + 6x + 6x + 9 + 5
= 4x2 + 12x +14
Domain = ldquoall realsrdquo
Composition of Functions Example 3
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(f(x))
f(2x + 3) = 2(2x + 3) + 3
= 4x + 6 + 3
= 4x + 9
Domain = ldquoall realsrdquo
One More Example
bull f(x) = 3x and g(x) = x14
bull f(x) g(x)
Domain = ldquopositive real srdquo
1
4
3x
x
3
43x
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
Composition of Functions Example 1
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(g(x))
f(x2 + 5) = 2(x2 + 5) + 3
= 2x2 + 10 + 3
= 2x2 + 13
Domain = ldquoall realsrdquo
Composition of Functions Example 2
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find g(f(x))
g(2x + 3) = (2x + 3)2 + 5
= (2x + 3)(2x + 3) + 5
= 4x2 + 6x + 6x + 9 + 5
= 4x2 + 12x +14
Domain = ldquoall realsrdquo
Composition of Functions Example 3
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(f(x))
f(2x + 3) = 2(2x + 3) + 3
= 4x + 6 + 3
= 4x + 9
Domain = ldquoall realsrdquo
One More Example
bull f(x) = 3x and g(x) = x14
bull f(x) g(x)
Domain = ldquopositive real srdquo
1
4
3x
x
3
43x
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
Composition of Functions Example 2
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find g(f(x))
g(2x + 3) = (2x + 3)2 + 5
= (2x + 3)(2x + 3) + 5
= 4x2 + 6x + 6x + 9 + 5
= 4x2 + 12x +14
Domain = ldquoall realsrdquo
Composition of Functions Example 3
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(f(x))
f(2x + 3) = 2(2x + 3) + 3
= 4x + 6 + 3
= 4x + 9
Domain = ldquoall realsrdquo
One More Example
bull f(x) = 3x and g(x) = x14
bull f(x) g(x)
Domain = ldquopositive real srdquo
1
4
3x
x
3
43x
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
Composition of Functions Example 3
bull f(x) = 2x + 3 and g(x) = x2 + 5
bull Find f(f(x))
f(2x + 3) = 2(2x + 3) + 3
= 4x + 6 + 3
= 4x + 9
Domain = ldquoall realsrdquo
One More Example
bull f(x) = 3x and g(x) = x14
bull f(x) g(x)
Domain = ldquopositive real srdquo
1
4
3x
x
3
43x
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
One More Example
bull f(x) = 3x and g(x) = x14
bull f(x) g(x)
Domain = ldquopositive real srdquo
1
4
3x
x
3
43x
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
Review and some more1 Domain look at beginning through ending
2 Canrsquot divide by zero
3 Canrsquot take even root of a negative number
bull Difference between ldquononnegativerdquo and ldquopositiverdquo real numbersndash Nonnegative Real Numbers
bull Includes ldquo0rdquo
ndash Positive Real Numbersbull Does NOT include ldquo0rdquo
1
2x 1
2
1
x
Which has nonnegativeand which has positivereal s for the domain
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
bull EX f(x) = 2x + 5 g(x) = x2 - 1
bull f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = ldquoall realsrdquo
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
Homework
P41828-31 all (answer questions amp evaluate
each for x=3)
40-51 all
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
73 Power Functions and Function Operations
Day 2
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
bull Go over- 28 - 41
- 30 - 44
- 31 - 47
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
19
Interval Notationbull Brackets mean gt or lt [ ]
bull Parenthesis mean gt or lt ( )
2x 0x
023 x
(2 )
( 0] [ 230)
( )
Interval Notation
5x ( 5) (5 )
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
Domain for 3 morehellip
Positive v Nonnegative
1 1x or
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
Homework
bull 73 WS Practice B2 - 16 even
17 - 22 all (answer question amp evaluate each problem for x=2)
ndash Write out domain for ALL in ldquoInterval Notationrdquo
