Math 3208
 Course Outline (From NLESD)
 You will need a Calculator
◦
◦
Graphing calculators are not required but they can be helpful
from view graphs.
Alternatively you can download apps on your phone for viewing
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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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 "f (x)" means ___________________________
__________________________________________;
◦
◦
the expression does _____ mean ______________________
Don't embarrass yourself by pronouncing (or thinking of) "f(x)"
as being "f times x".
 In function notation, the "x" in "f (x)" is called "the __________
of the function", or just ________________________.
◦
So if we give you "f (2)" 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 "minus" sign.
 Remember that "x" 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 _____
_________________ to worry about.
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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

x





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

domain.

What is the Domain



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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
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

D) Sketch y = f(x)
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Math 3208 Unit 1


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
Combination by Addition/Subtraction
 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


What type of
function is h(x)?

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

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x
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
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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:

y

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x
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

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Math 3208 Unit 1

 Graph the difference of these two graphs:

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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
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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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x
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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