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Let A = {x: -1 <= x <= 1} = B and consider mappings f: A -> B. For each function, state whether it is bijective or non-bijective. (a) f(x) = x|x| (b) f(x) = x³ (c) f(x) = sin(pi*x/2)

1. (a) bijective, (b) bijective, (c) bijective
2. (a) non-bijective, (b) bijective, (c) bijective
3. (a) bijective, (b) non-bijective, (c) bijective
4. (a) bijective, (b) bijective, (c) non-bijective

Correct answer: (a) bijective, (b) bijective, (c) bijective

Solution

(a) x|x| = x² for x>=0 and -x² for x<0; strictly increasing on [-1,1], maps [-1,1] onto [-1,1] -> bijective. (b) x³ strictly increasing, maps [-1,1] onto [-1,1] -> bijective. (c) sin(pi*x/2) strictly increasing on [-1,1] (argument ranges over [-pi/2, pi/2]), endpoints -1 and 1, onto [-1,1] -> bijective.

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