AAT
1

Simplify functions by adding, subtracting, multiply or dividing.
#1
Simplify a composition of functions.
#2
#3
#4
#5
#6
#7
#8
#9
#10
Graph functions using transformations
1 2 3 4 5
Write equations based on their transformation
1 2 3 4 5
Graph piecewise functions
1 2 3 4 5
Find function values for real life piecewise function situations
1 2 3 4 5
Multiply and simplify radical expressions
1 2 3 4 5
Simplify by adding, subtracting, multiplying and dividing radical expressions
1 2 3 4 5
Solve radical equations
1 2 3 4 5
Solve radical inequalities
1 2 3 4 5
2

Target 1:_________________________________________________________________________________________
Recall Function Operations:
 
 3 x  1 ( )
 x
2 
5 x

3 a. f(-3) = b. 3f(0)+25 =
Notations for operations on functions
Adding --
 f
 g
       c. g(-4) =
Subtracting -- d. 5g(1/3)
 f
 g
      
Multiplying --
 f
 g
      
Dividing --

 
   

0
Ex #1
If
  
3 x
2 
7 x and a.
 f
 g
 
Ex #2
If
  
3 x
2 
2 x and
  
2 x
2 x 1 find the following b.
 f
 g
 
  x 4 find the following c.
 f
 g
  a.
 f
 g
  b. f
  c.
 f
 g
 
3

Target 2:__________________________________________________________________________________________
Composition functions- combining functions…symbolized
 f
    
( )

read as “f of g of x”
Ex #3 Given a.
( ) x 3 and
 f
  
( )
 x
2 
1 , find the following b.
 g f
   c.

(2)

Ex #4 Given a.
( )

3 x
2  x and ( )

2 x

1 , find the following
 f
   b.
 g f
   c.

(2)

Relation- a set of ordered pairs (they share a relationship)
Ex #5
If
   and a.

(5)
 b.

(3)
 c.

(5)

  d.


(7)

4

Target 1 & 2 Homework
Use the functions
   x
2 
4 x
 
2 3
1.
 f
 g
 
2.
 f
 g
 
3.
 f
 g
 
4. f
 
Use the functions
9.
 f
  
Use the functions
12.
 
  
5 x

2
 
6
  x 3
 
2
2
13.
10.
 g f
  
 
5.
 f
 g
 
6.
 f
 g
 
7.
 f
 g
 
8. f
 
11.

( 2)

14.

(2)

5

Use the functions f

(6)

15.

( 4, 5),(0,3),(1,7)

&
16.

( 5)
 g
  
More Composition of functions……
Given f(x) = 2x + 3, g(x) = 3x – 4, and h(x) = 2x 2 – 3x + 4 find each function.
18) (f g)(x) 19) (f h)(x)
21) f(g(3)) 20) (h f)(x)
22) g(h(0)) 23) (f f)(x) for problems #15-17
17.

( 4)

6

Target #3__________________________________________________________________________________________
Graph each of these on your graphing calculator. These are the PARENT FUNCTIONS or basic shapes for the graphs of each type of function.
Quadratic Cubic Absolute Value Square Root
a.
( )
 x
2 b.
( )
 x
3 c.
( )
 x
d.  x
Transformation Rules
Rule #1: Graph each on your calculator and compare each one to the parent functions above.
( ) ( x 3)
2
( )

( x

1)
3 f x
 
5 ( )
 x

5
What do you notice about them? How did they change or move?
____________________________________________________________________________
Rule #2 Graph each on your calculator and compare each one to the parent functions above.
( )
 x
2 
4 ( )
 x
3 
2 ( ) x 5 ( )
 x

2
What do you notice about them? How did they change or move?
____________________________________________________________________________
Rule #3 Graph each on your calculator and compare to the parent function.
  x
2   x
3   x
What do you notice about them? How did they change or move?
  x
____________________________________________________________________________
Combined Rules: Graph each on your calculator and compare each one to the parent graph.
7

a)
( ) x 3 2
b)   x
For a-c, describe the change in comparison to their parent function.
c) x 3
___________________ ____________________ ____________________
___________________ ____________________ ____________________
Sketch the graph by hand using the transformation rules, then check your graph in the calculator.
1) f x

( x

1)
2 
3
2) x 2
3) x 2 4
4)  x 3 1
Investigate a few new transformations: a. Given the parent function ( )
 x , sketch the ( )

4 x on your calculator and explain the change. b. Given the parent function ( )
 x
2 ,
sketch the function

1 x 2 on your calc
4 and explain the transformation.
8

d) c. Given the parent function  x
, sketch the function
( )

3 x
on your calculator and explain the transformation
that took place.
Sketch the graph by hand using the transformation rules, then check your graph in the calculator.
5) x 4
6) f x ( x 1)
3 
5
7)
( )

2 x

3
8) x 5 a)
Target #4______________________________________________________________________
Identify the parent function and then write the equation. Check your equation on the calculator. b) c)
_________________ ___________________ ___________________
_________________ e)
__________________ f)
__________________
9

Target 3 & 4 Homework
Sketch the graphs by hand using the transformation rules, then check your work on the graphing calculator.
1)
 x 5 3 2) f x ( x 4)
3
3) x 1 6
4) f x

3 x
2 
4 5)

1
4 1
3 x 6) x 2
Write the equation for the graph shown below. State the domain & range of the graph also.
7) 8) 9)
Equation:_____________________ Equation:_____________________ Equation:______________________
Domain:_____________________
Range:______________________
Domain:______________________ Domain:________________________
Range:________________________ Range:_________________________
10) 11) 12)
Equation:_____________________ Equation:_____________________ Equation:______________________
Domain:_____________________ Domain:______________________ Domain:________________________
Range:______________________ Range:________________________ Range:_________________________
10

Target 5:_________________________________________________________________________________________ x  4 if x  2
1.
1 if x  2
D:________________
R:________________ f(-1)=
3.

 
 x  1 if x
2 x if x
  1
  1
D:________________________
R:________________________ h(0)=
5.

 x

 x
2
2
2 if x if
 
  x
3
  1 if x  1
j(-2)=
2. g(3)=
D:__________________
R:__________________ f(1)=
 
 x 1 if x
( x  3) 2  3
 3 if x  3
D:___________________
R:___________________
 
 x 2 3 if x
1 if x  0
 0
6.
( )
 ( x  4) 2


 if x   4
1
2
1 x  2 if x if

  x 
2
2 f(-4)=
4.
11

Target 6:___________________________________________________________________________________________
Ex 1: A company gave us a quote for prom t-shirts based on the following piecewise function. There is a $10 charge to create the silk screen that will be put on the front of the t-shirt. In the piecewise function below number of t-shirts and ( ) represents the cost . x represents the
( )




 x
0
10 3.50 , x
100
350

350 a. How much would it cost for 85 t-shirts?__________________________________ b. How much would it cost for 350 t-shirts?__________________________________ c. How much would it cost for 556 t-shirts?__________________________________
Ex 2: SokoCell Company has different text messaging plans. If you text less than 250 messages a month your cost is a basic fee of $5 plus $0.07 per text. If you text between 250 and 1000 messages a month then your cost is a basic fee of
$5 plus $0.05 per text. If you text over 1000 messages per month then your cost is a flat fee of $60. Write a piecewise function for the cost, c(x), based on the number of text messages sent in a month.
 







 a. How much would it cost for 753 text messages?_________________________ b. How much would it cost for 2200 messages?___________________________
Ex 3: In the summer, Jack works as a lifeguard at the neighborhood pool. He earns $8.50 per hour up to his regular 40 hour work week. Any hours he works past 40 hours he earns time and a half. Jack is not allowed to work more than 65 hours in a week. The piecewise function below shows Jack’s earnings.
8.50 , 0 40
 x

40), 40 65 a. How much would Jack earn if he works 54 hours in a week?_______________________ b. If Jack earned $212.50, how many hours did he work?______________________________
12

Target 5 & 6 Homework
1. x
2
3 if x if x
 
  1
1
D:
R: f(-3)=
3.
R:
D: f(0)=
 

(
3 x  1) 2  if x
2
 if x
0
 0
5.
 
 x 2 if x x 3 if x  0
 0
D:
R: f(-2)=
R: f(2)=
2:
( )  
 x
4 if x if x 

3
4
D:
4:
( )



1
2 x x

 
4
3
D:
R: f(5)= if x if x


2
2
6:
( ) 





2
4 x x
 1 if x if
  1
  x  if x  3
3
D:
R: f(0)=
13

7. A parking garage charges money based on the number of hours you park in the garage. The piecewise function below shows the cost, c(x), based on the number of hours parked in the garage, x.
( )



2,

0 x
20, 7
1
24
7 a. How much would it cost if you parked in the garage for 45 minutes?_________________________ b. How much would it cost if you parked for 7 hours?___________________________
8. Two cell phone companies costs are described in the piecewise functions below. Each function shows the cost c(x) based on the number of minutes used in a month, x.
Grunloh Mobil



50, 300
0
750
300 de Groh Cellular
( )




0

55, 500 x 200
800
200
500 a. Which plan is better for you to choose if you usually talk 450 minutes a month?______________________________
Why? Show your math here….. b. Which plan is better for you to choose if you talk 300 minutes a month?____________________________________
Why? c. Which plan costs less if you use 150 minutes per month?_______________________________________________
Why?
14

Target 4- 6 Practice - Graph each function. State the domain & range.
1. y  
1
2 x 2  4
2. y  



1
2 x x  3,
, x x


0
0
Domain:_____________________
Range:_______________________
Domain:____________________________
Range:_____________________________
3.
( ) 
 x  2
 3
 5
4.

 
 x x
2

 3,
1 , x x


0
0
Domain:_____________________
Range:_______________________
Domain:____________________________
Range:_____________________________
5.

 3
2 x
6.
( )  
 x
3 ,
 2, x x
  2
  2
Domain:_____________________
Range:_______________________
Domain:____________________________
Range:_____________________________
15

7. y 



(
2 x x


1

4) if x
2 x


3

4
0 if x  4
8. y  x 3 x  1




2   x  x  5
 3
 1
1 x   4
Domain:_____________________
Range:_______________________
9. Write a piecewise function for the real life example below.
Domain:____________________________
Range:_____________________________
SGD – Tunes offers a pricing schedule based on how many songs a person downloads from their website. The cost for downloading less than 20 songs is $1.99 per song. The cost for downloading between 20 and 50 songs is $1.50 per song plus a basic fee of $3. The cost for downloading between 51 and 100 songs is $0.99 per song plus a fee of $5.
( )
 







 a. How much does it cost to download 30 songs?______________________________________ b. How much does it cost to download 51 songs?_____________________________________
16

Review Targets 1-6
1. Target 1: Given
  x g x a.
 h
 g
  d.
 f
 g
 
 x 2  b.
 f
 g
  e. h
 
 x 2  4 x  1 find c.
 h
 f
  f.
 f
 g
 
2. Target 2: Given
 x  x a.
 f
  
3. Target 3: Graph the following: b.
 a. f x  x  3) 2  1 x 2 x 1 find
 b. g x   ( x  2) 3  3 g f
  
c.
( ) x 1 c.

(2)
 d.

1
3 x  2
17

4. Target 4: Write the equation for the graph shown below. State the domain & range of the graph also. a. b. c.
Equation:_____________________ Equation:_____________________ Equation:______________________
Domain:_____________________ Domain:______________________ Domain:________________________
Range:______________________ Range:________________________ Range:_________________________
5. Target 5: Graph the piece-wise functions and state the domain and range. a.
( )



2
3, x  3, x
3, 1 x 
 1
 x 
4
4 b.


 x x x


1,
3, x
4 1, f
Domain: ______________________
Range: _______________________
  f  f 
 0 x
2 x  0
Domain: ______________________
Range: _______________________ g   g 
  3 g  
6. Target 6: Write a piecewise function for the real life example.
The local circus charges $10 per ticket. If you buy more than 25 tickets at once, they only charge $7.50 a person.
Construct a piecewise function that represents the ticket prices, and find how much it would cost for a group of 25 people to go to the circus.
18

1)
Target 7:___________________________________________________________________________________________
To simplify square roots_____________you need to make groups of ___________________________
To simplify cube roots _____________you need to make groups of _____________________________
To simplify fourth roots____________you need to make groups of _____________________________
Examples:
6 5
28 a b
2)
3 6 12
8 a b
3)
4
4 9
16 x y
5)
27 xy z
6)
4
5 12 x y
7)
4)
3  15 9
27 a b
32 x y 10
8)
3 12 10
64 a b
9)
4 16 9 a b
19

To Multiply Radicals__________________________________________________________________________________
Examples:
2)
4 5 a
5 
125 a
3
1)
5 2 x

3 8 x
4)
4 3
3 2 x y

4 27 xy
2
5)
3)
3 36 xy

2 6
2 2 x y x y 2  x y 2
6)
2 32
3 5 a b

8
7 2 a b
20

Target 7 Homework:
Simplify
1)
3 27 a b 3
2)
4 625 a b 5
3)
4
5 12
32 x y
10)
4)
20 a b c
7)
3 a 4 a
6 
6 a
2
3 4 x y
 2 7 x y
5)
3 32 x y z
8)
3 27 xy

3 36
3 2 x y
11)
6)
5 5 a b c 6
9)
4 4
4 6 x y
4 12
13 50 ab a b

4 8
3 2 x y

6 8
18 2 ab a b
21

Target 8:_______________________________________________________________________
Example 1: Simplify. What is the operation?____________________________________________
To add or subtract radicals you must have like _______________
Steps:1.__________________________ 2.____________________________ a) 5 32

27

2 75

3 b) 4 40

3 28

200
Example 2: Simplify. What is the operation?________________________________________
To multiply binomial radicals ______________________________________________ a)
 

4 5
 b)

8 3



2 2

22

Target 8 Homework:
Simplify completely.
1)
6 20

8 5 5 45
3)
  
5)

2 3 7

4

7

7) Find the perimeter & area of the rectangle
.
2)
8 48 6 75

7 80
4)
Area:
2 3

15

60

6)
4 2
 x y
3
Perimeter:
23

Target 9:___________________________________________________________________________________________
Steps: 1._______________________________________________________________________
2._______________________________________________________________________
3._______________________________________________________________________
***________________________________________________________
Example 1: Solve
2) 21  5 x   3)   x
  1) x 2 4 7
4)  x

3
 1
3
4 0
5)  x

6
 1
4

12

0 6)
4 2 x   
24

Target 10:______________________________________________________________________
**___________________________________________________________________
Example 2: Solve.
1) 3

5 x

10

8 2) 6 x
4) 2 x 2 1 5 3) 2  4 x  
25

Target 9 & 10 Homework
26

11)
Test Review
Target 1 & 2: Perform the function operation.
Use the functions
1) ( g

)( )
( )
 x
2 
5 , g x

4 x

5 &
2) ( f

)( )

1

3
2 x
4) ( h g )( 4) 5)
 f g

( ) 6) ( ( ))
3) ( f

)( )
7) ( h f )(3)
Target 3: Graph Functions using transformations.
8)
 x 2 1 9) f x ( x 2)
3 
3 10) ( ) x 3 4
( )
 x
2 
1 12) ( )
 
2 x 2 3 13)
  x

2
27

Target 4: Write equations based on their transformations.
14) 15) 16)
Equation:_____________________ Equation:_____________________ Equation:______________________
Domain:_____________________
Range:______________________
Target 5: Graph Piecewise Functions.
17)
2 x

2 x

1 if if x x

1
 
1
Domain:______________________ Domain:________________________
Range:________________________ Range:_________________________
Domain:______________________
Range:________________________ f(-5)=____ f( 2)=______
18)
 
 x

2 if x
( x

3)
2
 
2 if x

1
Domain:________________________
Range:_________________________ f(-5)=_____ f(0)=____
19) ( )




2 x

3
| |
3 x 3 if x
 
3 if if x

2 x 2
Domain:____________________________
Range:_____________________________ f(-300)=_____ f(1)=_____
28

Target 6: Find function values of real life piecewise function situations.
20) The average newborn’s resting heart rate is 112 bpm and continues to decrease by 2 bpm until the age of 20. After
20 a person’s resting heart rate averages 72 bpm. Write a piecewise defined function for the function. What is your average resting heart rate?
Target 7: Multiply and Simplify Radicals
22)
4 81
23)
3 16w v
21)
64x 6
24)
3 49m t
25)
5  32x y
26)
27)
4 2 xy 2

4 8 x y 3
Target 8: Add, subtract, multiply and divide radical expressions.
28)
12 2 3  108
29)



29

30)

2 

 2

Target 9: Solve radical equations.
32) x
34) Solve.
3 x   x  4
Target 10: Solve radical inequalities.
36) Solve. x
31)

2  6
2 
33) Solve.
3 2 w  4
35) Solve.
 t  3 
1
3
 2
37) Solve.
3 2 x  
30