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Name: ____________________ Algebra II Honors
Pre-Chapter 3 Homework
Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc…
Try and do as many as you can without a calculator!!!
x a
n
The “nth root of a”
n
a a
1
n
Be able to think backwards….. Evaluate the Radicals below!
n=2
(perfect
squares)
n=3
(perfect
cubes)
n=4
n=5
n=6
x a
x a
x a
x a
x a
16 
3
2
4
5
6
1 
1 
1 
1 
16 
22 
23 
24 
25 
26 
32 
33 
34 
35 
36 
42 
43 
44 
45 
46 
52 
53 
54 
55 
62 
63 
64 
72 
73 
82 
83 
2
3
92 
93 
10 
10 
2
11 
.
12 2 
.
132 
.
2
14 2 
152 
.
.
.
3
4
5
.
49 
81 
196 
3
8
3
27 
3
729 
3
1000 
4
16 
4
81 
4
256 
5
32 
6
64 
.
.
.
.
.
.
10 
.
.
10 4 
105 
6
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Section 3.1:
Evaluate nth Roots and Use Rational Exponents
The “nth root of a”
n
a a
1
n
for any integer n greater than 1. n is the ______________ of the radical.
a is called the_______________ and can be be a real number.
Rational Exponents
Let
a
a
a
m
n
m
1
n
be an nth root of a, and let m be a positive integer.
1
 (a n ) m  ( n a ) m
n
1

1

(a n ) m
Example 1:
Evaluate expressions with rational exponents.
2
a. 125
1
Example 3:

3
Example 2:
a. 22
1
,a  0
m
n
( a)
b.
8
4 3

Approximate with a calculator.
4

b. 35
5
6

c. ( 5 11)4 
Solve Equations using nth roots.
n is odd
n is even
a. x5  32
b. x5  32
c. x6  1
d. x6  1
3
e. 6 x  384
f. ( x  8)5  100
g. ( x  5)4  16
h.
1 8
x 0
3
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Let n represent a positive integer.

Under what conditions will an equation of the form
xn  a

Under what conditions will an equation of the form
x n  a have exactly one real solution?

Under what conditions will an equation of the form
x n  a have two real solutions?
have no real solutions?
Section 3.2: Apply Properties of Rational Exponents
Recall the properties of exponents:
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67
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Section 3.6: Solve Radical Equations
Example 1
Example 3
Example 5
2
5
( x  7)
4
Example 2
Example 4
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Equations with Extraneous Solutions
Extraneous solutions are solutions that emerge from the process of solving an equation, but are not a
valid solution to the context of the original problem.
Example 1
x  2  2 x  28
Example 2
10  b  2  3b  2
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Section 3.5: Graph Square Root and Cube Root Functions
Graph: f ( x)  3 x
Graph: y  x
Domain: _____________
Range: _____________
Domain: _____________
Range: ____________
Transformations take functions and “transform” them through vertical/horizontal shifts and reflections.
An easy way to understand transformations is to think of functions as having two parts; an “inside” and an
“outside”. Most of the time, “inside” refers to the part of the equation inside parentheses or under symbols.
“Inside” transformations move functions horizontally while “outside” transformations move functions
vertically. Transformations “inside” are always the opposite from what is given to you and transformations
“outside” are taken as they are given.
Let’s look at an example of a general transformation.
f ( x)  a x  h  k or f ( x)  a 3 x  h  k , etc.
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Here is what each part does: Let’s look at the “outside” first.
a→ coefficient
* if a is between 0 and 1, it is considered a vertical compression.
* if a is larger than 1, it is considered a vertical stretch.
* if a is negative, it is a reflection over the x-axis.
k→ constant: gives you a vertical shift up/down.
*if it is+𝑘, it will move up that many
*if it is−𝑘 , it will move down that many
Now let’s look at the “inside”. Remember that “inside” transformations are the opposite of what appears.
h→ constant: gives you a horizontal shift left/right.
*if it is + h, it will move left that many
*if it is - h, it will move right that many
Graph Radical Functions:
Example 1:
Graph x  4  2
a) State the name of the function: _________________________________
b) State the transformation:
____________________________________________________________________________________
c) Graph:
Original Function
Stretch/Compression
Vert/Hor
/Reflection
Shifts
d) Domain:_______________
Range:_______________ (Interval Notation)
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Example 2:
Graph 3 3 x  8  6
a) State the name of the function: _________________________________
b) State the transformation:
____________________________________________________________________________________
c) Graph:
Original Function
Stretch/Compression
Vert/Hor
/Reflection
Shifts
d) Domain:_______________
Range:_______________ (Interval Notation)
Example 3:
Graph 2 x  1  1
a) State the name of the function: _________________________________
b) State the transformation:
____________________________________________________________________________________
c) Graph:
Original Function
Stretch/Compression
Vert/Hor
/Reflection
Shifts
d) Domain:_______________
Range:_______________ (Interval Notation)
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BIG IDEA ABOUT DOMAIN!!!
Assume when you are given a function that the domain will be all real numbers or  ,   unless you see one of the
following conditions: If any of these conditions exist in the function, the domain has to be “restricted”.
1. Variables in the denominator of a fraction: Set denominator  0 and solve.
2. Variables underneath even radicals or a base raised to a rational exponent with an even denominator:
Set underneath or base  0 and solve.
3. If there are variables underneath even radicals or a base raised to a rational exponent with an even
denominator that are in the denominator of a fraction:
Set underneath or base  0 and solve since the denominator cannot equal zero!
Find the domain and write your answer using interval notation.
1.
f ( x)  3 x  x 2  1
Domain: ____________________
2.
1
3
f ( x)  x  2 x
1
4
Domain: ____________________
3.
g ( x) 
2 x2
2x 1
Domain: ____________________
4. g ( x) 
3x  5
7
Domain: ____________________
36 x  2  2x
5. h( x) 
x 5
Domain: ____________________
6. h( x) 
4
x2
Domain: ____________________
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Section 3.3: Perform Function Operations and Composition
Notation:
The functions 𝑓 + 𝑔, 𝑓 − 𝑔, 𝑎𝑛𝑑 𝑓𝑔 are defined for each x for which both 𝑓 and 𝑔 are defined. That is to say,
the domain of arithmetic combinations is the intersection of the domains of 𝑓 and 𝑔.
(𝑑𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑓 + 𝑔) = (𝑑𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑓 − 𝑔) = (𝑑𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑓𝑔) = 𝐷𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑓 ∩ 𝐷𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑔
𝑓
The function 𝑔 is defined for each x for which both 𝑓 and 𝑔 are defined and 𝑔(𝑥) ≠ 0.
𝑓
That is,(𝑑𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑔) = 𝐷𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑓 ∩ 𝐷𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑔, as long as 𝑔(𝑥) ≠ 0
Perform the following function operations on the following functions. State the domain of each using
interval notation.
a) f ( x)  2 x and g ( x)  x  1
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b)
f ( x)  2 x  1 and g ( x) 
1
x 3
Domain of f:
Domain of g:
Domain of
intersection:
c)
f ( x)  x 2 and g ( x)  x  1
Domain of f:
Domain of g:
Domain of
intersection:
Example 1: Add and subtract functions
Let
f ( x)  5 x
1
3
and g ( x)  11x
a. f ( x)  g ( x)
1
3
. Find the following.
f ( x )  g ( x)
b.
c. the domains of f  g and f  g
Example 2: Multiply and divide functions
Let f ( x)  8 x and
a.
f ( x)  g ( x)
5
g ( x)  2 x 6 . Find the following.
b.
f ( x)
g ( x)
c. the domains of
f g
and
f
g
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Examples of notation for composition of functions.
Rules for Composition of functions:
Whatever is last function in line is the first machine. You can think of it as working from the “inside, out.”

Ex. 𝑓(𝑔(𝑥)) would set up like this:
𝑔(𝑓(𝑥)) would set up like this:
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2
Example 1: Let f ( x)  3x  4 and g ( x)  x  1 . What is the value of f ( g (3)) ?
Finding the domain of a composite function consists of two steps:
Step 1. Find the domain of the "inside" (input) function. Graph the domain of the inside function.
Step 2. Construct the composite function. Find the domain of this new function. Graph the domain of the
composite function. Use the most restrictive domain (or the intersection of the domains). The composite may
result in a domain not necessarily equal to the domains of the original functions.
Example 2:
Given the following functions, find the function composition and state the domain in interval notation.
f ( x)  2 x 1
g ( x)  x 2  1
f ( g ( x))  ____________________
a)
1
h( x )  3 x 2  x 2
1
p ( x)  x 2
j ( x)  x 2  5
b) ( g f )( x)  ____________________
b)
c)
d)
Domain of g:
Domain of f:
Domain of f:
Domain of g:
Domain of
Intersection
Domain of
intersection:
Domain: ____________________
c) f f  ____________________
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
q)
r)
s)
t)
u)
v)
w)
x)
y)
z)
aa)
bb)
Domain: ____________________
d) g ( j ( x))  ____________________
‘
Domain: ____________________
Domain: ____________________
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Practice: Given the following functions, find the function operation or composition and state the domain in
interval notation.
f ( x)  2 x 1
g ( x)  x 2  1
e) h( x)  p( x)  ____________________
g)
i)
k)
h( x )  3 x  x
2
1
2
p ( x)  x
1
2
j ( x)  x 2  5
f) h( x)  g ( x)  ____________________
Domain: ____________________
Domain: ____________________
( p g )( x)  ____________________
h) h( x) p( x)  ____________________
Domain: ____________________
Domain: ____________________
g ( p( x))  ____________________
j) j p  ____________________
Domain: ____________________
Domain: ____________________
j ( x)
 ____________________
g ( x)
Domain: ____________________
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Section 3.4: Use Inverse Functions
One-to-one functions: A function where neither the x-value nor the y-value repeat.
The test for a one-to-one function is the horizontal line test. Like the vertical line test used to determine if a
relation is a function, you draw a horizontal line through the graph and if it crosses once, it is one-to-one. If it
crosses more than once, it is not one-to-one.
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Inverse Functions: In order for a function to have an inverse that is also a function, the original function must
be one-to-one. The inverse of a one-one function is obtained by switching the role of x and y, then re-solve
the equation for y. The new function will be identified by the notation 𝑓 −1 (𝑥).
Properties of inverse functions:




The domain of 𝑓 −1 = the range of 𝑓.
The range of 𝑓 −1 = the domain of 𝑓.
The inverse of 𝑓 −1 is 𝑓, that is (𝑓 −1 )−1 = 𝑓.
Inverse functions are symmetric about the line 𝑦 = 𝑥.
Example 1:
Find the inverse of f ( x)  x 2  3
Graph of f.
Domain of f:_______________
Range of f:_______________
One-to-one? Y
N
Inverse is:_________________
Graph of inverse.
Domain of inverse:_______________
Range of inverse:_______________
Is our inverse an inverse function? Y
N
If not, what can we do to make our inverse an inverse function?
Write the inverse function using the correct notation._______________________
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Example 2:
Find the inverse of f ( x)  3x  4
Graph of f.
Domain of f:_______________
Range of f:_______________
One-to-one? Y
N
Inverse is:_________________
Graph of inverse.
Domain of inverse:_______________
Range of inverse:_______________
Is our inverse an inverse function? Y
N
If not, what can we do to make our inverse an inverse function?
Write the inverse function using the correct notation._______________________
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Example 3:
Find the inverse of f ( x)  x 2  2, x  0
Graph of f.
Domain of f:_______________
Range of f:_______________
One-to-one? Y
N
Inverse is:_________________
Graph of inverse.
Domain of inverse:_______________
Range of inverse:_______________
Is our inverse an inverse function? Y
N
If not, what can we do to make our inverse an inverse function?
Write the inverse function using the correct notation._______________________
3𝑥 + 5
5
𝑎𝑛𝑑 𝑔(𝑥) =
𝑎𝑟𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑠
𝑥
𝑥−3
−1
−1
𝑢𝑠𝑖𝑛𝑔 (𝑓 ∘ 𝑓 )(𝑥) & (𝑓 ∘ 𝑓)(𝑥).
𝐸𝑥𝑎𝑚𝑝𝑙𝑒 1: 𝑉𝑒𝑟𝑖𝑓𝑦 𝑡ℎ𝑎𝑡 𝑓(𝑥) =
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3
𝐸𝑥𝑎𝑚𝑝𝑙𝑒 2: 𝑉𝑒𝑟𝑖𝑓𝑦 𝑡ℎ𝑎𝑡 𝑓(𝑥) = −2𝑥 3 + 1 𝑎𝑛𝑑 𝑔(𝑥) = √
𝑢𝑠𝑖𝑛𝑔 (𝑓 ∘ 𝑓 −1 )(𝑥) & (𝑓 −1 ∘ 𝑓)(𝑥).
You Try…
−𝑥 + 1
𝑎𝑟𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑠
2
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Decomposition of a function
Sometimes we can write a given function as the composition of two or more other functions. This is called
decomposing the function. It is important to be able to decompose functions in later work in calculus.
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For example, take the function ℎ(𝑥) = √2𝑥 + 1. For this function h, find functions f and g such that
ℎ(𝑥) = 𝑔(𝑓(𝑥)).
3
In this case, we could write 𝑓(𝑥) = 2𝑥 + 1 𝑎𝑛𝑑 𝑔(𝑥) = √𝑥.
 The rule can change depending on the question being asked, for example, if the directions say that
3
ℎ(𝑥) = 𝑓(𝑔(𝑥)) and then 𝑔(𝑥) = 2𝑥 + 1 𝑎𝑛𝑑 𝑓(𝑥) = √𝑥.
For the given function h, find functions f and g such that ℎ(𝑥) = 𝑓(𝑔(𝑥)).
Example 1: ℎ(𝑥) = (𝑥 + 1)2 − 3(𝑥 + 1) + 4
𝑓(𝑥) =_______________
𝑔(𝑥) =_______________
For the given function h, find functions f and g such that ℎ(𝑥) = 𝑔(𝑓(𝑥)).
Example 2: ℎ(𝑥) = (𝑥 3 + 1)2
𝑓(𝑥) =_______________
𝑔(𝑥) =_______________