Warm-up Score
Name:
Note Packet Score
Algebra 2: UNIT 11
Inverse and Radicals
Warm-up #1
Score _________
11.1: Inverse Functions and Relations
Goal 1: Find the inverse of a function or relation.
Goal 2: Determine whether two functions are inverses.
Inverse Relation:
1. Algebraically:
2. Graphically:
Example 1: Find the inverse Relation of the order pair: G={(3,4),(2,8)(1,3)}
Domain of G:
Domain of inverse:
Range of G:
Range of the inverse:
Example 2: Find the inverse Relation of the order pair: G={(-2,5),(5,5)(4,-6)}
Domain of G:
Domain of inverse:
Range of G:
Range of the inverse:
Inverse Functions:
Functions f(x) and g(x) are inverses of each other if and only if: f ( g ( x))  g ( f ( x))  x The function g is denoted by
f 1 , read as “f inverse”.
1
3
Example 3: f ( x)  3 x  6 and g ( x)   x  2 . Are 𝑓(𝑥) and 𝑔(𝑥) inverse functions?
3
4
Example 4: 𝑓(𝑥) = 4 𝑥 − 6 and 𝑔(𝑥) = 3 𝑥 + 8. Are 𝑓(𝑥) and 𝑔(𝑥) inverse functions?
Example 6: Find the inverse of the function f ( x) 
2
x  2 and graph f(x) and f 1 ( x) .
3
y




x









Example 7: Find the inverses of the following functions.
a. f ( x) 
1
x2
4
b. g ( x)  3  x
c. h( x) 
Example 8: Determine whether each pair of functions are inverse functions.
f ( x)  4 x  1
a.
1
g ( x)  (1  x)
4
Homework: page 395: 10, 13, 14, 19-27, 30-35
f ( x)  13x  13
b.
1
g ( x)  x  1
13
x4
3




Warm-up #2
Score _________
11.2: Square Root Functions and Inequalities
Goal 1: Graph and analyze square root functions.
Goal 2: Graph square root functions.
GRAPHS OF RADICAL FUNCTIONS (SQUARE ROOTS)
Graph Parent Graph
y
x
To graph y  a x  h  k , follow these steps:
Step 1:
Step 2:
Example 1: Graph y 
x  3 . State the domain and range.
Example 2: Graph y  2 x  4 . State the domain and range.
Example 3: Graph y  
1
x  3  2 . State the domain and range.
2
Example 4: Graph y 
x  4  1 . State the domain.
Example 5: Graph y 
1
x  1  2 . State the domain.
2
Homework: Worksheet (HINT: QUIZ next class)
Warm-up #3
Score _________
7.4: nth Roots
Goal 1: Simplify radicals.
Goal 2: Use a calculator to approximate radicals.
The inverse of raising a number to the nth power is finding the nth root of a number. To simplify a radical, “make a
tree”. Look for numbers that appear _____ times.
EVEN ROOTS:
ODD ROOTS:
Example 1: Simplify the following:
a.
f.
4
 343x 9 y 3
25x 4
b.  4 81x12
c.
( a + b )16
g.  5  32
h.  3 27
3
i.
15 20
e.  5 32 x y
(5) 2
d.
4
4 2
j.  100p q
16w 4v 8
Example 2: Use a calculator to approximate each value to three decimal places.
a.
304
Homework: Worksheet
b.
3
 490
c.  5 236
d.
6
(723)3
Warm-up #3
Score _________
7.5 Operations with Radical Expressions
Goal 1: Simplify radical expressions.
Goal 2: Add, subtract, multiply, and divide radical expressions.
Property
Rule
n
Product Property
Quotient Property
Example
ab  n a  n b
n
a na

b nb
A radical expression is simplified when the following conditions are met:
o The index n is as small as possible.
o The radicand contains no factors other than 1 that are nth powers of an integer or polynomial.
o EXAMPLE: 16 is not simplified because “4” is a factor that appears twice.
o There are no radicals in the denominator.
Example 1: Simplify
a.
3
54
b.
16 p 8 q 7
c.
4
64 x 4 y 6
d.
x4
y5
e.
5
5
4a
Multiplying and Adding Radicals YOU CAN ONLY MULTIPLY RADICALS IF THE INDEX IS THE SAME
Same Index: 63 9a 2  33 24a
Difference Indexes:
3x  3 5 x
Example 2: Simplify.
a.  2 15  4 21
b. ( 34 24 )(54 20 )
Adding and Subtracting Radicals YOU CAN ONLY ADD RADICALS IF THE INDEX AND THE NUMBER INSIDE THE
SQUARE ROOT ARE THE SAME
Same Index & Number: 2 17  3 17
Different Index & Number: 4 3  2 27
Example 3: Simplify
a.
2 12  3 27  2 48
b. (3 5  2 3 )( 2  3 )
Example 4: Simplify (Hint: Use a conjugate to rationalize a denominator)
1 3
5 3
Homework: Worksheet
Warm-up #4
Score _________
7.6: Rational Exponents
Goal 1: Write expressions with rational exponents in radical form and vice versa.
Goal 2: Simplify expressions in exponential or radical form.
RADICAL FORM:
EXPONENTIAL FORM:
Example 1: Evaluate
a. 16

1
4
b. 243
1
4 2 2
3
5
c. (16 g h )
Example 2: Simplify each expression
2
1
6
7
6
a. x  x  x
b. y

3
4
c.
18rs 3
1
6r 4 t 3
Example 3: Simplify each expression
8
81
a.
6
3
Homework:
b.
4
9z
2
c.
m
m
1
1
2
1
2
1
Warm-up #5
Score _________
7.7 Solving Radical Equations and Inequalities
Goal 1: Solve equations containing radicals.
Goal 2: Solve inequalities containing radicals.
Power property of equality: If a  b , then a  b
n
n
GETTING RID OF A RADICAL:
x2  9
Example 1: Solve
x 9
2x  8  4  6
Example 2: Solve 5  4 x  0
x 3  27
3
Example 4: Solve
x 5
4 x  28  3 2 x  0
Example 5: Solve x  2  2 x  8
1
4
3
Example 3: Solve 3x  243
3
Example 6: Solve 3(5n  1)  2  0