Math 106 Fall 2011 Function vs. Process FUNCTION TYPES CONCEPTUAL UNDERSTANDINGS (PROCESSES) A.    Polynomial (a-linear, b-quadratic, c-cubic, and d-higher degree forms) 1.      Solving: solving equations, finding zeros, taking inverses  (Use reverse Order of Operations) B.     Rational 2.      Evaluating: evaluating functions, evaluating expressions, finding y-intercepts (Follow Order of Operations) C.     Piecewise 3.      Recognizing characteristics of functions: finding intercepts, finding asymptotes, end-behavior, maxima and minima D.    Exponential 4.      Translating from the symbolic to the graphic representation of a function E.     Logarithmic 5.      Translating from the graphic to the symbolic representation of a function 6.      Finding domain and range 7.      Finding the inverse of a composite or combined function 8.      Combining Functions: algebraic combinations, compositions, and decompositions 9.      Modeling

For each function type (Aa through E) listed, be able to demonstrate each of the processes (1 through 9) listed.

A Four Step Problem Solving Process*:

While every problem is different, the process of problem solving can share common features. The following four-step process provides a useful guide to problem solving.  Keep in mind that these four steps offer general advice and do not automatically lead to a solution.  This method is offered as a good starting point that is, a way to proceed if you are unsure as to what you should do first.

STEP I:  Understand the Problem.  Be sure that you understand the nature of

the problem.  For example:

·   Think about the context of the problem (that is, how it relates to other problems in the real world) to gain insight into its purpose.

·   Make a list or table of the specific information given in the problem, including units for numerical data.

·   Draw a picture or diagram to help you make sense of the problem.

·   Restate the problem in different ways to clarify its question.

·   Make a mental or written model of the solution, into which you can insert details as you work through the problem.

·   Make sure you understand ALL of the terminology used and how the context affects the facts.

Step II:  Devise a strategy for solving the problem.  Finding an appropriate

strategy requires creativity, organization, and experience.  In seeking a strategy, try any or all of the following (as well as others you may think of):

·   Obtain needed information that is not provided in the problem statement, using recall, estimation or research.

·   Make a list of possible strategies and hints that will help you select your overall strategy.

·   Map out your strategy with a flow chart or diagram.

·   Recall the formulas or strategies with which you are already familiar and think about how they might relate to the current situation.

·   Recall if you have solved a similarly stated problem in the past.

Step III:  Carry out your strategy, and revise it if necessary.  In this step you

are likely to use analytical and computational tools.  As you work through the mathematical details of the problem, be sure to do the following:

·   Keep an organized, neat, and written record of your work, which will be helpful if you later need to review your solution.

·   Double each step so that you do not risk carrying or minus sign errors through to the end of your solution.

·   Constantly reevaluate your strategy as you work.  If you find a flaw in your strategy, return to Step II and create a revised strategy.

Step IV:  Look back to check, interpret, and explain your result.  Although you

may be tempted to think you have finished after you find a result in Step III, this final step is the most important.  After all, a result is useless if it is wrong or misinterpreted or cannot be explained to others.  Always do the following:

·   Be sure that your result makes sense.  For example, be sure that it has the expected units, that its numerical value is sensible, and that it is a reasonable answer to the original problem.

·   Once you are sure that your result is reasonable, recheck your calculations or find an independent way of checking the result.

·   Identify and understand potential sources of uncertainty in your result.

·   Write your solution clearly and concisely, including discussion of any relevant uncertainties or assumptions.

·   Consider and discuss any pertinent implications of your result.