Transcript Non-Algebraic Functions - Appalachian State University

Algebraic vs. NonAlgebraic Functions
Jeneva Moseley
Department of Mathematics
University of Tennessee
[email protected]
Why do we call a function a
machine?
What is a “non-algebraic function”?
Teacher: What is a non-algebraic function?
Smart-aleck student: A function that is not
algebraic.
Teacher: OK. Then what is an algebraic
function?
Definitions of Algebraic Functions:
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“functions which can be formed by these
operations: addition, multiplication,
division, and the nth root”
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“a function which satisfies a polynomial
equation whose coefficients are
themselves polynomials”
Develop a “Field Guide for
Functions.”
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Algebraic ones: Linear, quadratic, power,
polynomial, rational
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Non-algebraic ones:
http://webgraphing.com/examples_transc
endentals.jsp
Algebraic Structure of
a Group
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g ∘ f, the composition of f and g. For example, (g ∘ f )(c) = #.
A composite function represents the application of one function
to the results of another. For instance, the functions f : X → Y and
g : Y → Z can be composed by first computing f(x) and then
applying a function g to the output of f(x).
Thus one obtains a function g ∘ f : X → Z defined by
(g ∘ f )(x) = g (f (x)) for all x in X.
The composition of functions is always associative. That is, if f, g,
and h are three functions with suitably chosen domains and
codomains, then f ∘ (g ∘ h) = (f ∘ g) ∘ h.
The functions g and f are said to commute with each other if
g ∘ f = f ∘ g. In general, composition of functions will not be
commutative. Commutativity is a special property, attained only by
particular functions. But a function always commutes with its
inverse to produce the identity mapping.
Inverse Functions:
Tricks of the Trade
Reflections over y-axis:
What reflections and shifts will we
recognize?
Geometric Definitions
of the Trig Functions
θ
Geometric Definitions
of the Trig Functions
θ
Using similar triangles and Pythagorean
Theorem, derive some trig identities…
What if theta is obtuse?
θ
With rational functions, it might be
helpful to know long division for
polynomials.
x  12 x  42
3
2
 x  3 x  12 x  0 x  42
x 3
3
2
Examples of creating models:
Until 1850, humans used so little crude oil
that we can call the amount zero.
 By 1960, humans had used a total of 600
billion cubic meters of oil.
 Create a linear model that describes world
oil use since 1850. Discuss the validity of
this model.
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y=mt + b
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Validity:
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Parameters vs. Correlation Coefficients:
What kind of parameters does this
have?
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The number of hours of daylight varies with the
seasons. Use the following data for 40 degrees N
latitude (of San Francisco, Denver, and D.C.) to
model the change in the number of daylight hours
with time.
– The number of hours of daylight is greatest on the
summer solstice (June 21), when it is 14 hrs.
– The number of hours of daylight is smallest on the winter
solstice (Dec 21), when it is 10 hrs.
– On the spring and fall equinoxes (Mar 21, Sept 21), there
are 12 hours of daylight.
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According to the model, at what times of the year
does the number of daylight hours change most
gradually? Most quickly? Discuss the validity of the
model.
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Consider an antibiotic that has a half-life in
the bloodstream of 12 hours. A 10milligram injection of the antibiotic is
given at 1:00 pm. How much antibiotic
remains in the blood at 9:00 pm? Draw a
graph that shows the amount of antibiotic
remaining as the drug is eliminated by the
body.
The rule of three, I mean four.
Graphical, numerical, algebraic, verbal
 http://www.wmueller.com/precalculus/fam
ilies/1.html
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Families of Functions and Ideas for
Modeling each one
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Linear Functions:
http://www.wmueller.com/precalculus/families/1_10.html
Exponential Functions:
http://www.wmueller.com/precalculus/families/1_20.html
Logarithmic Functions:
http://www.wmueller.com/precalculus/families/1_30.html
Power Functions:
http://www.wmueller.com/precalculus/families/1_40.html
Polynomial Functions:
http://www.wmueller.com/precalculus/families/1_50.html
Rational Functions:
http://www.wmueller.com/precalculus/families/1_60.html
Trigonometric Functions:
http://www.wmueller.com/precalculus/families/1_70.html