Transcript Power Functions and Function Operations

7.3 Power Functions and
Function Operations
1)
2)
Goals
Perform operations with functions, including power functions.
Use power functions and function operations to solver real-life
problems.
Evaluate each function for when x = -2
• f(x) = 2x + 5
• g(x) = x2 – 1
• f(x) = 2(-2) + 5
• f(x) = -4 + 5
• f(x) = 1
• g(x) = (-2)2 – 1
• g(x) = 4 -1
• g(x) = 3
Domain
• The numbers that CAN be plugged in
for x in a function
– Have to look at beginning function through
the ending function for domain!!!
• Try to find the values that CANNOT
be plugged in to determine domain
1. Can’t divide by zero (undefined)
2. Can’t take even root of a negative
number
• If there are no restrictions on x, then
the domain is “all reals”
3
Operations with Functions
• The four basic operations can be applied
to any function.
• Let’s perform the basic operations with this
example: f(x) = 2x + 5
g(x) = x2 - 1
Addition of Functions
• EX: f(x) = 2x + 5
g(x) = x2 - 1
• f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = “all reals”
Is f(x) + g(x) = g(x) + f(x)?
Subtraction of Functions
• EX: f(x) = 2x + 5
g(x) = x2 - 1
• f(x) – g(x) = (2x + 5) – (x2 - 1)
= 2x + 5 – x2 + 1
= -x2 + 2x + 6
Domain = “all reals”
Is f(x) – g(x) = g(x) – f(x)?
Multiplication of Functions
• EX: f(x) = 2x + 5
g(x) = x2 - 1
• f(x) · g(x) = (2x + 5)(x2 - 1)
·
3
= 2x - 2x + 5x2 - 5
= 2x3 +5x2 - 2x - 5
Domain = “all reals”
Is f(x) · g(x) = g(x) · f(x)?
Division of Functions
• EX: f(x) = 2x + 5
g(x) = x2 - 1
• f(x) / g(x) = (2x + 5) / (x2 - 1)
2x  5
 2
x 1
·
Domain = x  1 or  1
Is f(x) / g(x) = g(x) / f(x)?
Composition
• Plug one function into another
–Put one function in place of x
h( x)  f ( g ( x))
9
Composition of Functions:
Example 1
• f(x) = 2x + 3 and g(x) = x2 + 5
• Find: f(g(x))
f(x2 + 5) = 2(x2 + 5) + 3
= 2x2 + 10 + 3
= 2x2 + 13
Domain = “all reals”
Composition of Functions:
Example 2
• f(x) = 2x + 3 and g(x) = x2 + 5
• 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 = “all reals”
Composition of Functions:
Example 3
• f(x) = 2x + 3 and g(x) = x2 + 5
• Find: f(f(x))
f(2x + 3) = 2(2x + 3) + 3
= 4x + 6 + 3
= 4x + 9
Domain = “all reals”
One More Example
• f(x) = 3x and g(x) = x1/4
• f(x) / g(x)

3x
x
1
4
·
 3x
Domain = “positive real #s”
3
4
Review and some more!
1. Domain: look at beginning through ending
2. Can’t divide by zero
3. Can’t take even root of a negative number
• Difference between “nonnegative” and
“positive” real numbers
Which has nonnegative
– Nonnegative Real Numbers
• Includes “0”
– Positive Real Numbers
• Does NOT include “0”
and which has positive
real # s for the domain ?
1
1
2

x
x
1
2
• EX: f(x) = 2x + 5
g(x) = x2 - 1
• f(x) + g(x) = (2x + 5) + (x2 - 1)
= 2x + 5 + x2 - 1
= x2 + 2x + 4
Domain = “all reals”
EVALUATE for x = -4
= x2 + 2x + 4
= (-4)2 + 2(-4) + 4
= 16 - 8 + 4
= 12
Homework
P.418
#28-31 all (answer questions & evaluate
each for x=3)
#40-51 all
7.3 Power Functions and
Function Operations
Day 2
• Go over:
- 28
- 30
- 31
- 41
- 44
- 47
Interval Notation
• Brackets mean > or < [ ]
• Parenthesis mean > or < ( )
Interval Notation
(2, )
x2
x0
(, 0]
23  x  0
[23, 0)
( ,  )
x5
19
(,5)  (5, )
Domain for 3 more…..
Positive v. Nonnegative
x  1 or  1
Homework
• 7.3 WS Practice B
#2 - 16 even
#17 - 22 all (answer question & evaluate each
problem for x=2)
– Write out domain for ALL in “Interval Notation”