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Algebraic vs. Non- Algebraic Functions Jeneva Moseley Department of Mathematics University of Tennessee jmoseley@math.utk.edu

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Why do we call a function a machine?

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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?

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Definitions of Algebraic Functions: “functions which can be formed by these operations: addition, multiplication, division, and the nth root” “functions which can be formed by these operations: addition, multiplication, division, and the nth root” “a function which satisfies a polynomial equation whose coefficients are themselves polynomials” “a function which satisfies a polynomial equation whose coefficients are themselves polynomials”

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Develop a “Field Guide for Functions.” Algebraic ones: Linear, quadratic, power, polynomial, rational Algebraic ones: Linear, quadratic, power, polynomial, rational Non-algebraic ones: http://webgraphing.com/examples_transc endentals.jsp Non-algebraic ones: http://webgraphing.com/examples_transc endentals.jsp http://webgraphing.com/examples_transc endentals.jsp http://webgraphing.com/examples_transc endentals.jsp

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Algebraic Structure of a Group g ∘ f, the composition of f and g. For example, (g ∘ f )(c) = #. 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). 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. 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 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. 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.

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Inverse Functions: Tricks of the Trade

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Reflections over y-axis:

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What reflections and shifts will we recognize?

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Geometric Definitions of the Trig Functions θ

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θ Using similar triangles and Pythagorean Theorem, derive some trig identities…

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What if theta is obtuse? θ

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With rational functions, it might be helpful to know long division for polynomials.

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Examples of creating models: Until 1850, humans used so little crude oil that we can call the amount zero. 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. 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. Create a linear model that describes world oil use since 1850. Discuss the validity of this model.

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y=mt + b y=mt + b Validity: Validity: Parameters vs. Correlation Coefficients: Parameters vs. Correlation Coefficients:

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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 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. 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. 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 10- milligram 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. Consider an antibiotic that has a half-life in the bloodstream of 12 hours. A 10- milligram 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.

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The rule of three, I mean four. Graphical, numerical, algebraic, verbal Graphical, numerical, algebraic, verbal http://www.wmueller.com/precalculus/fa milies/1.html http://www.wmueller.com/precalculus/fa milies/1.html http://www.wmueller.com/precalculus/fa milies/1.html http://www.wmueller.com/precalculus/fa milies/1.html

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Families of Functions and Ideas for Modeling each one Linear Functions: http://www.wmueller.com/precalculus/families/1_10.html Linear Functions: http://www.wmueller.com/precalculus/families/1_10.html http://www.wmueller.com/precalculus/families/1_10.html Exponential Functions: http://www.wmueller.com/precalculus/families/1_20.html Exponential Functions: http://www.wmueller.com/precalculus/families/1_20.html http://www.wmueller.com/precalculus/families/1_20.html Logarithmic Functions: http://www.wmueller.com/precalculus/families/1_30.html Logarithmic Functions: http://www.wmueller.com/precalculus/families/1_30.html http://www.wmueller.com/precalculus/families/1_30.html Power Functions: http://www.wmueller.com/precalculus/families/1_40.html Power Functions: http://www.wmueller.com/precalculus/families/1_40.html http://www.wmueller.com/precalculus/families/1_40.html Polynomial Functions: http://www.wmueller.com/precalculus/families/1_50.html Polynomial Functions: http://www.wmueller.com/precalculus/families/1_50.html http://www.wmueller.com/precalculus/families/1_50.html Rational Functions: http://www.wmueller.com/precalculus/families/1_60.html Rational Functions: http://www.wmueller.com/precalculus/families/1_60.html http://www.wmueller.com/precalculus/families/1_60.html Trigonometric Functions: http://www.wmueller.com/precalculus/families/1_70.html Trigonometric Functions: http://www.wmueller.com/precalculus/families/1_70.html http://www.wmueller.com/precalculus/families/1_70.html

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