You have given a function λ : R → R with the following properties (x ∈ R, n ∈ N): λ(n) = 0 , λ(x + 1) = λ(x) , λ(n+1/2)=1

Find two functions p, q : R → R with q(x)6=0 for all x such that λ(x) = q(x)(p(x) + 1).

One possible combination:

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Step-by-step explanation:

The requirement that for all suggests that should be a cyclic function with a period of .

One such example is the sine function. Let denote a non-zero real number. Consider the expression .

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Note how replacing with does not change the value of . Therefore, the period of would be .

The question requires that replacing the with should not change the value of . In other words, the period of should be . If is indeed in the form , the value of should ensure that . Therefore, .

It would take a few more transformations to ensure that for all integers , and .

One possible approach is to make each a trough of this function, and each a crest of this function. The corresponding sine wave would have a range of only . However, without any transformation, would have a range of two.

Therefore, it would be necessary to compress vertically by a factor of , so that its range meets the needs.

is the output of one such compression and has the correct period and range. However, its troughs would be at (rather than ) whereas its crests are at (rather than .) It would be necessary to shift this function upwards by unit for the troughs and crests to meet match the ones in the requirements.

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answer: 1) twenty-seven times y

              2) twenty-seven multiplied by y