Chapter
1 - Functions
2 - Limits and Continuity
3 - Differentiation
4 - Applications of Derivatives
5 - Integration
6 - Applications of Definite Integrals
CHAPTER 1
Functions
1.1 Functions and Their Graphs
1.2 Combining Functions;
Shifting and Scaling Graphs
1.3 Trigonometric Functions
1.1
Functions and Their Graphs
1.1 Functions and Their Graphs
Definition
Domain and Range
Graphs of Functions
Piecewise –Defined Functions
Increasing and Decreasing Functions
Even and Odd Functions
Common Functions
It is a function, because:
Every element in X is related to Y
No element in X has two or more relationships
It is a relationship, but it is not a function, because:
Value "3" in X has no relation in Y
Value "4" in X has no relation in Y
Value "5" is related to more than one value in Y
Domain = All possible input values (or) x
values
Range = All possible output values (or) y
values
Finding Domain and Range
Find the domain of the following functions (support graphically):
f  x  x  3
The key question: Is there anything that x could not be???
x3 0
x  3
Always write your answer in interval notation:
D :  3,  
Finding Domain and Range
Find the domain of the following functions (support graphically):
x
g  x 
x 5
What are the restrictions on x ?
x 5  0
x0
x5
Interval notation:
D :  0,5 
 5,  
Graphs of Functions
Graphs of Functions
f x   C
Graphs of Functions
f x   x
Graphs of Functions
f x   x
Graphs of Functions
f x   x
3
The Vertical Line Test for a Function
A function f can have only one value f (x) for each in its domain, so no
vertical line can intersect the graph of a function more than once. If a
is in the domain of the function f, then the vertical line x  a will
intersect the graph of at the single point (a, f (a)) .
This is a function
This is NOT a function
Greatest Integer Function and Least Integer Function
Increasing and Decreasing Functions
On a given interval, if the graph of a function rises from left to right, it is said to
be increasing.
If the graph drops from left to right, it is said to be decreasing.
 If the graph stays the same from left to right, it is said to be constant
Even – symmetric about the y-axis, ex) y = x2
Odd – symmetric about the origin, ex) y = x3
Ex. 1
Even, Odd or Neither?
Graphically
f ( x)  x
Algebraically
f ( x)  x
f (1)  1  1
f (1)  1  1
They are the same, so it is.....
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27
Ex. 2
Even, Odd or Neither?
f ( x)  x  x
Graphically
3
Algebraically
What
happens if
we plug in 1?
f ( x)  x  x
3
f (2)  (2)  (2) 
6
3
f (2)  (2)  (2)
3
 6
They are
opposite, so…
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28
Ex. 3
Even, Odd or Neither?
Graphically
f ( x)  x  1
2
Algebraically
f ( x)  x  1
2
f (1)  (1)  1
2
2
2
f (1)  (1)  1
2
They are the same, so.....
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29
Ex. 4
Even, Odd or Neither?
f ( x)  x  1
Graphically
3
Algebraically
f ( x)  x  1
3
f (1)  (1)  1
3
0
3
f (1)  (1)  1
 2
They are not = or opposite, so...
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30
for constants m and b
p and q are polynomial