Calculus 1.2

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1.2 day 1 Functions and Graphs
The Parthenon, Centennial Park, Nashville,
Tennessee
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Athena Statue (holding Nike), Centennial Park,
Nashville, Tennessee
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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
pre-calculus 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
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Dont 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 dont have to stress
over passing your first college math class!
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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.
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Example
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
or
This is set builder notation.
We usually use interval notation.
These are open intervals (with parentheses)
because we dont count the zero. Also notice the
symbol for union.
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or
Looking at the graph, we see that we can have
every y value except zero, so the range is
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Another example
Since we cannot take the square root of a
negative number, the domain is
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.
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Another example
Since we cannot take the square root of a
negative number, the domain is
Looking at the graph, we see that there are no
negative y values, so the range is
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Symmetry
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.
Any polynomial function with only even exponents
behaves the same way, and has y-axis symmetry.
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Any function with y-axis symmetry is called an
even function.
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A polynomial function with only odd exponents
has origin symmetry.
Changing the sign of x changes the sign of y.
In other words, if (x,y) is on the graph, so is
(-x,-y).
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Any function with origin symmetry is called an
odd function.
Polynomial functions with exponents that are both
even and odd have no symmetry.
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Of course, a graph with x-axis symmetry is not a
function at all!
Fails the vertical line test!
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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
Just graph each piece separately, for each part
of the domain.
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Example
Write a piecewise function for the graph at right
There are two pieces to this graph, so we need
two equations.
Paying attention to the open and closed circles,
we get the piecewise function
p