By Joshua Chover
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Consider the following two properties of functions: (i) f{a-\-b) = f{a) +f{b) for all rt a n d i n the domain. (ii) /(rt • b) = J \a ) • f{b) for all a and b in the domain. For each function defined below, determine whether it has either of these properties. (a) Sample:fix) =2x-\~S for all real x. ii)f{a + b) =2(fl + ¿^)+3 = 2fl + 2¿>+ 3 1 \ not equal f M - h f i b ) = (2fl + 3) + (2¿^ + 3) =2fl + 2¿>+ 6 J (ii)/(at) = 2ab + 3 1not equal for all f M f ( b ) = (2i/-f3) (2/^ + 3) i rt, in the domain = 4ab + + 6« + 9j For example, \eta = b = \ /(1 • 1) = 5 /(1)/(1) = 25 (b) J(x) = 0 for all real x (c) Jix) = I for all real X (d) fi x) = —4x for all real x (e) Jix) = X for all realx (0 f i x) = Vxforx ^ 0 (g) fi x) = x^ for all real x *(h) f ix) = logioxforx > 0 14.
Dss h ^ 1 8 ^ 8 ^ 8 ^ 1 8 ddd ^ 8 sds dss .. dds ddd * 4 8 ^ 8 etc. EXAMPLE 2: A disease with 2-to-l chances of survival for an untreated victim. In this case it seems reasonable to assign twice as much elementary probability to a pattern such as dsd as to the pattern ddd— since the middle victim’s chances are independent of the fates of the other two. Similarly, let’s assign twice as much elementary probability to ssd as to dsdy and twice as much to sss as to ssd. Let’s denote the elementary probability of ddd by w.
Since y ^ 1, 0 1“ (1/)’) < and this X can be used as an input yielding (b) All real numbers (c) The set of numbers 0 ^ ^ 1 f i x) = 1 f i x) = 2x —3 (d) The set of four numbers -2 ,-1 ,0 , 2 (e) The set of positive integers f ix) = |jc| (f) The set of numbers 0, i, S, f, 1 f ix) = 2 i x + 1) (g) All negative numbers fi x) = x^ f i x) -- (h) All positive numbers *(i) All real numbers f i x) = V ? - |x| *(j) All real numbers fM=x^+l 16. Describe functions by formula or by “mapping diagram” (as in Problem 1 above) whose domains and ranges are as given below: (a) Sample: Domain: the set of three numbers 1, 2, 3; and Range: —2, —4, —6.