MATH 115 ACTIVITY 2:
MATH 115 ACTIVITY 2: Functions and their properties
WHY: "Function" is one of the most basic concepts for the study of calculus. The function idea is the basic mathematical form of the idea that the values of one quantity determine the values of another, so our work on "How do changes relate?" will mostly be carried out using the language and symbolism of functions.
LEARNING OBJECTIVES:
1. Know what is (and what isn't) a function and recognize domains and ranges.
2. Be able to work with function notation - including evaluation (with numbers and symbols), arithmetic and composition of functions.
3. Work as a team, using the team roles.
CRITERIA:
1. Success in completing the exercises
2. Success in having all members of the team understand all the material
3. Success in working as a team and in filling the team roles.
RESOURCES:
1. Your text: Section 1.3 (pp31 - 40)
2. Your graphing calculators
3. The team role desk markers (handed out in class for use during the semester)
4. 40 minutes
PLAN:
1. Select roles, if you have not already done so, and decide how you will carry out steps 2 and 3 (5 minutes)
2. Complete the exercises below, making sure that everyone understands and agrees with the results that are recorded. (30 minutes)
3. Assess the team's work and roles performances and prepare the Reflector's and Recorder's reports including team grade (5 minutes).
4 The spokesperson will (orally) answer some questions on your results.
DISCUSSION:
Your text gives a good presentation of the basic notions associated with a function
EXERCISES:
1. a.) For which of the following is there a function that determines y for each value of x? Give the domain of each of the functions (but not of the non-functions)
b.) For which of the following is there a function that determines x for each value of y? Give the domain of each of these functions (but not of the non-functions)
i.)
|x |3 |5 |6 |7 |9 |11 |13 |
|y |5 |2 |8 |3 |8 |2 |6 |
ii.)[pic]
iii.)
2. Give the domain (according to the agreement on domains) for each of these:
a.) [pic]
b.) [pic]
3. Given these three functions [pic], with the usual assumption on domains.
Calculate [Some will give symbolic answers - with x's or h's, etc., but others will give numerical results]
a.) [pic]
b.) [pic]
c.) [pic]
d.) [pic]
e.) [pic] [This is a difference quotient It gives the slope of the line through (3,g(3)) and (5,g(5)). It does not give the slope of g because g is not linear and does not have one slope]
4. The rule given below does define a function p of x (for each value of x there is only one value of p) with domain
-2 ≤ x < 8.
a.) Calculate: p(-1), p(3.5), p(4)
b.) Sketch the graph of y = p(x) [Because different pieces of the domain require different formulas, the graph is in pieces]
[pic]
SKILL EXER CISES
(Hand in next Thursday) Text p.40 (section 1.3) # 1-8, 10, 12, 28-31, 42, 46, 50-52, 61, 65, 70, 71
CRITICAL THINKING QUESTIONS:(answer individually in your journal)
1. Give three functions (as relations between quantities, not as formulas – that is, describe some relationships between quantities that change) that might be of interest in the study of life sciences or medicine.
2. For a linear function, we have a “slope” which gives the rate of change of y (or of the function – same idea) with respect to x . What kind of problems would there be in calculating the slope for a function that is not linear – such as f(x) = x2 ?
3. Why might piecewise-defined functions (such as #4, with different formulas for different parts of the domain) be important in the study of biology (think of functions in which the input variable is time and the output is something like blood-sugar level)?
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