1.2 day 1 Functions
and Graphs
The Parthenon,
Centennial Park, Nashville, Tennessee
Photo by Vickie Kelly, 2006
Greg Kelly, Hanford High School, Richland, Washington
Athena Statue (holding Nike),
Centennial Park, Nashville, Tennessee
Photo by Vickie Kelly, 2006
Greg Kelly, Hanford High School, Richland, Washington
We will start our study of calculus by spending a few days
in a quick review of several pre-calculus topics.
Whether you are comfortable with all of these precalculus topics or have forgotten many of them and have
to learn them now, it is okay either way.
The most important thing to remember in your study of
calculus is this:

Don’t panic!
Calculus is a challenging class for everyone.
If you ask questions, keep up with homework, and study for
quizzes and tests, this will be the best math class you have
ever taken, and you will be proud of your success.
and …
College will be way more fun if you have taken calculus in
high school and don’t have to stress over passing your first
college math class!

Domain and Range
Functions have an independent variable (often x) and
a dependent variable (often y).
In this discussion we are going to use x for the
independent variable and y for the dependent variable,
but we could use other letters.
domain: The set of all possible x values for a function.
range: The set of all possible y values for a function.
We can specify the domain of a function, or we can look
at the nature of the function to determine the natural
domain of the function.

Example:
1
Find the domain and range of: f  x  
x
When finding the domain, it is often easier to think of what
values that we cannot use for x.
In this case, x cannot equal zero. The domain is:
D   x x  0
or
D   ,0   0,  
This is set builder notation.
We usually use interval notation.
These are open intervals (with parentheses) because we
don’t count the zero. Also notice the symbol for union.

D   x x  0
1
f  x 
x
D   ,0   0,  
or

y

Looking at the graph, we
see that we can have every
y value except zero, so the
range is:



x
R   ,0   0,  
















Another example:
Find the domain and range of:
y x
Since we cannot take the square root
of a negative number, the domain is:
D  [0, )
This is a half-open interval.
We use a square bracket on the left boundary point
because we count the zero. (The interval is closed at zero.)
When infinity is a boundary, it is always an open interval on
that end, with parentheses.

Another example:
y x
Find the domain and range of:
Since we cannot take the square root
of a negative number, the domain is:
D  [0, )

Looking at the graph, we
see that there are no
negative y values, so the
range is:
y




x
R  0,  
















Symmetry
2
y

x
When we graph the function
, we see that it has
y-axis symmetry.
When we square the x, any
negative sign cancels out, so
changing the sign of x does not
change the y value.

y



Any polynomial function
with only even

exponents behaves the
same way, and has yaxis symmetry.

x




y  x4  2x2 1
This is
0
x , which has an even exponent.











Any function with y-axis symmetry is called an even function.
For example, y 
an even function.
So is y  cos

x is
y



 x .

x
















A polynomial function with only odd exponents has
origin symmetry.
Changing the sign of x changes
the sign of y.

y
 2, 4 

In other words, if (x,y) is on the
graph, so is (-x,-y).



x

1 3
y x
2











 2, 4 




Any function with origin symmetry is called an odd function.
For example, y  sin
is an odd function.
 x

y




x

Polynomial functions
with exponents that are
both even and odd have
no symmetry.















Of course, a graph with x-axis symmetry is not a function
at all!

y




x y
x













Fails the vertical line test!



Piecewise Functions
While many functions can be defined by a single formula,
others are defined by applying different formulas to
different parts of their domains.

Example:

 x, x  0

 2
y  f  x  x , 0  x  1

1,
x 1

y

Just graph each piece
separately, for each part
of the domain.
x
















Example:
y
Write a piecewise function for
the graph at right:
There are two pieces to this
graph, so we need two
equations.
The equation for the left hand
piece is: y  x
1,1
 2,1

x


For the right hand piece, the
equation is: y  0  1 x  1
This simplifies to: y  x  1
Paying attention to the open and
0  x 1
 x,
closed circles, we get the piecewise f  x   
x  1, 1  x  2

function:
p