Algebraic vs. Non-Algebraic vs. Non-Algebraic FunctionsAlgebraic Functions

Jeneva MoseleyJeneva MoseleyDepartment of MathematicsDepartment of Mathematics

University of TennesseeUniversity of Tennesseejmoseley@math.utk.edujmoseley@math.utk.edu

Why do we call a function a Why do we call a function a machine?machine?

What is a “non-algebraic What is a “non-algebraic function”?function”?

Teacher: What is a non-algebraic Teacher: What is a non-algebraic function?function?

Smart-aleck student: A function that is Smart-aleck student: A function that is not algebraic.not algebraic.

Teacher: OK. Then what is an algebraic Teacher: OK. Then what is an algebraic function?function?

Definitions of Algebraic Definitions of Algebraic Functions:Functions:

““functions which can be formed by functions which can be formed by these operations: addition, these operations: addition, multiplication, division, and the nth multiplication, division, and the nth root”root”

““a function which satisfies a a function which satisfies a polynomial equation whose polynomial equation whose coefficients are themselves coefficients are themselves polynomials”polynomials”

Develop a “Field Guide for Develop a “Field Guide for Functions.”Functions.”

Algebraic ones: Linear, quadratic, Algebraic ones: Linear, quadratic, power, polynomial, rationalpower, polynomial, rational

Non-algebraic ones: Non-algebraic ones: http://webgraphing.com/examples_trhttp://webgraphing.com/examples_transcendentals.jspanscendentals.jsp

Algebraic Structure Algebraic Structure of a Groupof a Group

gg ∘ ∘ ff, the , the compositioncomposition of of ff and and gg. For example, (. For example, (gg ∘ ∘ f f )(c) = )(c) = #.#.

A A composite functioncomposite function represents the application of one represents the application of one function to the results of another. For instance, the functions function to the results of another. For instance, the functions f f : : XX → → YY and and g g : : YY → → ZZ can be can be composedcomposed by first computing by first computing f(x)f(x) and then applying a function and then applying a function gg to the output of to the output of f(x)f(x)..

Thus one obtains a function Thus one obtains a function gg ∘ ∘ f f : : XX → → ZZ defined by defined by ( (gg ∘ ∘ f f )()(xx) = ) = g g ((f f ((xx)) for all )) for all xx in in XX. .

The composition of functions is always associative. That is, if The composition of functions is always associative. That is, if ff, , gg, and , and hh are three functions with suitably chosen domains are three functions with suitably chosen domains and codomains, then and codomains, then ff ∘ ( ∘ (gg ∘ ∘ hh) = () = (ff ∘ ∘ gg) ∘ ) ∘ hh. .

The functions The functions gg and and ff are said to commute with each other if are said to commute with each other if gg ∘ ∘ ff = = ff ∘ ∘ gg. In general, composition of functions will not . In general, composition of functions will not be commutative. Commutativity is a special property, be commutative. Commutativity is a special property, attained only by particular functions. But a function always attained only by particular functions. But a function always commutes with its inverse to produce the identity mapping.commutes with its inverse to produce the identity mapping.

Inverse Functions: Inverse Functions: Tricks of the TradeTricks of the Trade

Reflections over y-axis:Reflections over y-axis:

What reflections and shifts will What reflections and shifts will we recognize?we recognize?

Geometric Definitions Geometric Definitions of the Trig Functionsof the Trig Functions

θ

Geometric Definitions Geometric Definitions of the Trig Functionsof the Trig Functions

θ

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

What if theta is obtuse?What if theta is obtuse?

θ

With rational functions, it With rational functions, it might be helpful to know long might be helpful to know long

division for polynomials.division for polynomials.

3 23 212 42

3 12 0 423

x xx x x x

x

Examples of creating Examples of creating models:models:

Until 1850, humans used so little Until 1850, humans used so little crude oil that we can call the amount crude oil that we can call the amount zero.zero.

By 1960, humans had used a total of By 1960, humans had used a total of 600 billion cubic meters of oil. 600 billion cubic meters of oil.

Create a linear model that describes Create a linear model that describes world oil use since 1850. Discuss the world oil use since 1850. Discuss the validity of this model.validity of this model.

y=mt + by=mt + b

Validity: Validity:

Parameters vs. Correlation Parameters vs. Correlation Coefficients:Coefficients:

What kind of parameters does What kind of parameters does this have?this have?

The number of hours of daylight varies with The number of hours of daylight varies with the seasons. Use the following data for 40 the seasons. Use the following data for 40 degrees N latitude (of San Francisco, Denver, degrees N latitude (of San Francisco, Denver, and D.C.) to model the change in the number and D.C.) to model the change in the number of daylight hours with time.of daylight hours with time.– The number of hours of daylight is greatest on the The number of hours of daylight is greatest on the

summer solstice (June 21), when it is 14 hrs.summer solstice (June 21), when it is 14 hrs.– The number of hours of daylight is smallest on the The number of hours of daylight is smallest on the

winter solstice (Dec 21), when it is 10 hrs.winter solstice (Dec 21), when it is 10 hrs.– On the spring and fall equinoxes (Mar 21, Sept On the spring and fall equinoxes (Mar 21, Sept

21), there are 12 hours of daylight.21), there are 12 hours of daylight. According to the model, at what times of the According to the model, at what times of the

year does the number of daylight hours year does the number of daylight hours change most gradually? Most quickly? change most gradually? Most quickly? Discuss the validity of the model.Discuss the validity of the model.

Consider an antibiotic that has a half-Consider an antibiotic that has a half-life in the bloodstream of 12 hours. A life in the bloodstream of 12 hours. A 10-milligram injection of the 10-milligram injection of the antibiotic is given at 1:00 pm. How antibiotic is given at 1:00 pm. How much antibiotic remains in the blood much antibiotic remains in the blood at 9:00 pm? Draw a graph that shows at 9:00 pm? Draw a graph that shows the amount of antibiotic remaining as the amount of antibiotic remaining as the drug is eliminated by the body.the drug is eliminated by the body.

The rule of three, I mean The rule of three, I mean four.four.

Graphical, numerical, algebraic, Graphical, numerical, algebraic, verbalverbal

http://www.wmueller.com/precalculuhttp://www.wmueller.com/precalculus/families/1.htmls/families/1.html

Families of Functions and Ideas Families of Functions and Ideas for Modeling each onefor Modeling each one

Linear Functions: Linear Functions: http://www.wmueller.com/precalculus/families/1_10http://www.wmueller.com/precalculus/families/1_10.html.html

Exponential Functions: Exponential Functions: http://www.wmueller.com/precalculus/families/1_20http://www.wmueller.com/precalculus/families/1_20.html.html

Logarithmic Functions: Logarithmic Functions: http://www.wmueller.com/precalculus/families/1_30http://www.wmueller.com/precalculus/families/1_30.html.html

Power Functions: Power Functions: http://www.wmueller.com/precalculus/families/1_40http://www.wmueller.com/precalculus/families/1_40.html.html

Polynomial Functions: Polynomial Functions: http://www.wmueller.com/precalculus/families/1_50http://www.wmueller.com/precalculus/families/1_50.html.html

Rational Functions: Rational Functions: http://www.wmueller.com/precalculus/families/1_60http://www.wmueller.com/precalculus/families/1_60.html.html

Trigonometric Functions: Trigonometric Functions: http://www.wmueller.com/precalculus/families/1_70http://www.wmueller.com/precalculus/families/1_70.html.html