# Math 3208 unit 1 blank student notes 2019

```Math 3208
 Course Outline (From NLESD)
 You will need a Calculator
◦
◦
Graphing calculators are not required but they can be helpful
from view graphs.
graphs as you work through problems in class and at home.
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Math 3208 Unit 1
Unit 1
Pre-Calculus
Functions
1.1 _______________________________
1.2 _______________________________
Introduction
 In this section we’re going to make sure that you’re familiar with
functions and function notation. Both will appear in almost every
section in a Calculus class and so you will need to be able to deal
with them.
 We will work with the operations of functions. We will add,
subtract, multiply, and divide functions to create new functions.
 Finally we then progress to composite functions where we will
determine functional values and identify the domain and range of
the composite function.
What exactly is a function?
 An algebraic expression will be a function if for any __ in the
__________ of the expression (the domain is all the ___that can
be ______________ into the equation) the expression will yield
_________________ value of y.
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Math 3208 Unit 1
Example 1 Determine if each of the following are functions.
(a) y = x2 + 1
 This first one ________
___________.
 Given an x, there is ____
____ way to square it and
then add 1 to the result.
 So, no matter what value
of x you put into the
equation, there is ______
__________________.
(b) y2 = x + 1
 The only difference between
this equation and the first is
that we _______________
____________ off
the x and onto the y.
 This small change is all that
is required, in this case, to
change the equation from a
function to something that
_________________.
 To see that y2 = x+ 1 isn’t a function is fairly simple.
 Choose a value of x, say x=3 and plug this into the equation.
 Solve for y.
 Since there are ________________________ of y that we get
from a single x this equation __________________________.
How can we tell from a graph of an equation whether it is a
function or not?
 _______________________________
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Math 3208 Unit 1
 Sketch the previous 2 functions using technology
(a) y = x2 + 1
(b) y2 = x + 1
y
y
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x
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Next we need to take a quick look at function notation.
 Function notation is nothing more than a fancy way of
__________ the y in a function to something such as f(x).
 Let’s take a look at the following function:
 Using function notation we can write this as any of the following:
 Recall that this is NOT a letter ________ x, this is just a fancy
way of writing y.
So, why is this useful?
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Math 3208 Unit 1
 With this notation, you can now use _______________________
_______________ at a time without confusing yourself or mixing
up the formulas
 It provides an efficient way to __________________________.
◦
If we want to find the value of a function at a particular value,
we can just say _____________ rather than saying ________
_______________________________________________.
How do we go about evaluating functions?
 First, remember this: While parentheses have, up until now, always
indicated multiplication, the parentheses ___________________
______________________ in function notation.
◦
________________________________
 The expression &quot;f (x)&quot; means ___________________________
__________________________________________;
◦
◦
the expression does _____ mean ______________________
Don't embarrass yourself by pronouncing (or thinking of) &quot;f(x)&quot;
as being &quot;f times x&quot;.
 In function notation, the &quot;x&quot; in &quot;f (x)&quot; is called &quot;the __________
of the function&quot;, or just ________________________.
◦
So if we give you &quot;f (2)&quot; we are asking to ______ 2 into _____
________________________________________.
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Math 3208 Unit 1
Examples
 Given f (x) = x2 + 2x – 1, find f (2).
 Given g(x) = x3 + 2x2 – 1, find g(–3).
 NOTE: Using parentheses as we just did above helps keep track of
things like whether or not the exponent is on the &quot;minus&quot; sign.
 Remember that &quot;x&quot; is just a _____, waiting for something to be
put into it.
 Example: Given f(x) = -x2 +6x -11, find each of the following:
A) f(2)
B) f(-10)
c) f(t)
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Math 3208 Unit 1
Domain and Range
 The ___________ of a function is the set of all ____ that we can
plug into the function and get back a ______________ for y.
◦
At this point, that means that we need to ________________
________________ and taking _______________________
____________________________.
 The ___________ of a function is the set of all ____ that we can
ever get out of the function.

Determining the range of a function can be
_______________ to do for many functions.

This is where _______________________ comes in handy.
Determine the domain and range of the following
A) f(x) = x2 + 1
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Math 3208 Unit 1
B)
C)

In this case we’ve got a fraction, but notice that the
denominator will never be _______ for any real number since ___
is guaranteed to be ___________________________________
onto this will mean that the denominator is always ___________.
◦
In other words, the denominator won’t ever be ___________.
 So, all we need to do then is worry about the ________________
_______________________________.
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Math 3208 Unit 1
D)
 In this final part we’ve got both a _______________ and _____
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Math 3208 Unit 1
Piecewise Functions
A function that is defined by __________________________

The graph is drawn
y
by ______________

and _________ the
part that is _____ in

the restricted
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x
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domain.
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What is the Domain
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
and Range of the
Function?
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Math 3208 Unit 1
In math 2200 you used piecewise functions to describe Absolute
Value functions
EXAMPLE: Given the graph of : y  f (x )
a) sketch the graph
of y  f (x )
b) state the domain
and range
c) express y  f (x ) as a piecewise function.
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Math 3208 Unit 1
Other piecewise functions
2x  4, x  2
1. Sketch the function of: f (x )  
3  3x , x  2
y
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x
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State the domain and range
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Math 3208 Unit 1
x  2, x  2
2.A) Draw f (x )  
2
x  1, x  2
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B) Find Domain and Range
C) What value, instead of ____, would make this function have all
reals for the range?
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Math 3208 Unit 1
x 2  2, x  1
3. Consider: f (x )  
x  2, x  1
A) Find f(-1)
B) Find: f(10)
C) Find x when f(x) = 6
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D) Sketch y = f(x)
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Math 3208 Unit 1

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Practice: Page 9 and 10 #39, 41, 42
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Math 3208 Unit 1
1.1 Combining Functions
 Functions can be_____________________________________
______________________________ to form new relationships
between variables.
 In the last set of problems there were many illustrations of this.
 For example what combination of functions resulted in
Deﬁnition: sum, difference, product, quotient functions
 Suppose f and g are given functions. Functions denoted by f + g,
f – g, f g, and f are given by:
g
 Sum:
 Difference:
 Product:
 Quotient:
 The domain of each combined function is the set of all real
numbers for which the right side of the equation _____________
__________________________________________________
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Math 3208 Unit 1
 Consider functions:
f(x) = x2 + 8
g(x) = 6x
A) Complete the table
B) Plot y = h(x)
What is the
shape of h(x)?
y
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What type of
function is h(x)?
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x
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What is the
relationship
between the
domains of f(x),
g(x) and h(x)?

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Math 3208 Unit 1
C) Find h(x) algebraically
Vertex:
Intercepts:
D) How does the answer from part C compare to the graph in part
B)?
 How can we find the graph of either f(x) + g(x), (or f(x) – g(x)) if
we are only given the graphs of y = f(x) and y = g(x)?
◦
_______________________________________________
_______________________________________________
_______________________________________________
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Math 3208 Unit 1
 Graph the sum of these two graphs:
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y
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x
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Math 3208 Unit 1

 Graph the difference of these two graphs:
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y
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x
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Math 3208 Unit 1

 Complete the table of values below where:
h(x) = f(x) + g(x) and k(x) = f(x) – g(x)
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Math 3208 Unit 1
Examples: Find h(x) and its domain and range
A) h(x) = (f + g)(x) where f(x) = 2x and g(x) = 3x+2
B) h(x) = (f - g)(x) where f(x) = 2x2 -3x + 3 and g(x) = 3x2 -3x+2
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Math 3208 Unit 1
C) h(x) = (f - g)(x) where f (x )  x  1
and g(x) = 2
D) h(x) = (f +g)(x) where f(x) = |x – 1| and g(x) = 2
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Math 3208 Unit 1
Product and Quotient of Functions
 Determining the product and quotient of functions is similar to
finding the sum and product.
◦ However when finding the quotient the ____________ of
the combined function may have to be ____________ to
avoid _______________________.
Example 1:
 Consider f(x) = x + 4 and g(x) = x +1. Determine (fg)(x) and
(f/g)(x). State the domain of each combination.
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Math 3208 Unit 1
Example 2
 Consider f (x )  x and g (x )  x  1 .
A) Determine h(x) = f(x)g(x).
B) Determine s(x)=f(x)f(x). What is the domain of s(x)?
Example 3
2
 Consider f (x )  x  3x  4 andg (x )  x  1 .
A) Determine
h (x ) 
g (x )
f (x )
B) Check for non-permissible values for h(x)
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Math 3208 Unit 1
C) Use technology to sketch the graph of h(x).
y
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x
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D) What is occurring at the non-permissible values?
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Math 3208 Unit 1

Practice
Find f + g, f - g, f * g, and f / g for f(x) = x2 - 3x + 2, g(x) = x - 2
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Math 3208 Unit 1
Composition of Functions
Composition of Functions
•
Consider a function f(x) = sin x, and g(x) = 2x.
Find:
A) f(0)
B) g(0)
C) f( )
D) g(
)
E) f(g(x))
•
•
The _______________ of f(x) composed with g(x) is the function
________, or _____________
A composite function is a combination of functions where one
function is ____________ into another function.
EXAMPLE:
y = sin(3x +2)
•
•
This is a composite where _________ is inserted into the function
______.
3x + 2 is the _______ function and sin x is the ________
function.
EXAMPLE 1: Consider: f(x) =x2, g(x) =x + 1 . Find:
A) f  g (1) 
B) g f (1)
C)
f f (3)
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Math 3208 Unit 1
D) f(g(x))
E) g f (x )
F)
f f (x )
EXAMPLE 2:
1
f(x) = 2 – 3x, g(x) = x2, and h (x )  x  1
Find:
A) f g (x )
B) g f (x )
D) h(g(x))
E)
f
C) f f (x )
g h (x )
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Math 3208 Unit 1
Composition Using Graphs and Tables
Example: Use the graphs to find the following:
A) f(g(3))
B) g(f(-1))
C) f(f(2)
y
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Math 3208 Unit 1
Domain and Range of Composite Functions
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Math 3208 Unit 1
Note:
 The domain of (f (g(x)) is the set of elements x in the ________
_______ such that _____ is in the _________________.

The domain of the composite function f(g(x)) is found by taking
the ___________________ of the ________ of the inner
function, ____, and the __________of the outer function, ____
 The range of f(g(x)) is found by using the _______ of f(g(x)).
Example:
f (x )  x
Find f(g(x)) and its domain and range if: g (x )  x  1
What’s wrong?
When asked to find the domain of a composite function h(x) = (f (g(x))
where g(x) = x + 2 and f (x ) 
1
x
Billy wrote the following solution:
◦ The domain of g(x) is all real numbers and the domain of f
(x) is all real numbers where x ≠ 0. Combining these
restrictions will result in the domain of the composition, all
real numbers where x ≠ 0.
 Identify the error and write the correct solution.
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Math 3208 Unit 1
Decomposition
1. Find 2 functions that can be composed to give:
A) f(g(x)) = (x – 3)2
B) f(g(x)) = sin(x2)
C) f(g(x)) = sin2x
D )f ( g (x )) 
x 1
x 2
2. If f(x) = 3x +4 and f(g(x)) = 3x2 + 6x + 1, find g(x)
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Math 3208 Unit 1
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