1. Decide if each statement is True or False. If False,
1. Decide if each statement is True or False. If False, provide a
counterexample (stating domain D and the functions f and |f| involved; no explanation required). No proof of true statement requested.
(a.) _ _____ If f is continuous on D, then |f| is continuous on D.
(b) _ _____ If |f| is continuous on D, then f is continuous on D.
#4. For all parts, just state your answers; no explanation required. (You will need to do work to determine the answers, but you are not required to show it.)
Define for all real numbers x ¹ -2.
(a) To get a feel for how the function behaves, determine each of the following numerical values. That is, substitute a given x-value, simplify as appropriate, and then find the limit as n ® ¥.
f (-1) = _____ f (-1/8) = _____ f (0) = _____ f (1/2) = _____ f (2) = _____
f (-5/4) = _____ f (-3/2) = _____ f (-5/2) = _____ f (-3) = _____ f (- 4) = _____
(b) Determine the numerical value of f(x) for each real number x ¹ -2 —– you should find a relatively simple multi-part formula for f, with domain {x | x ¹ -2}. Just state your formula.
HINT: It can be helpful to plot the points you determined in part (a) to get a sense of the pattern.
(c) For what values of x is f continuous? (no explanation required)
#5. Let f: D ® R be continuous.
Decide if each statement is True or False. If False, provide a counterexample, stating D , f and f(D);
no explanation required. Note that your function f must be continuous. No proof of true statement requested.
(a) ________ If D is bounded, then f(D) is closed.
(b) ________ If D is closed, then f(D) is bounded.
(c) _________ If D is compact, then f(D) is compact. (Recall that compact = closed and bounded.)
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