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Unformatted text preview: MAS 111 FOUNDATION OF MATHEMATICS PRACTICE EXERCISES FUNCTIONS — Hints 1. Does the formula f ( x ) = 1 x 2 2 define a function f : R→ R ? HInt : No. f ( √ 2) and f ( √ 2) are not defined. 2. For each of the following functions, determine whether it is onetoone and determine its range. (a) f : Z→ Z , f ( x ) = 2 x + 1; (b) g : Q→ Q , g ( x ) = 2 x + 1; (c) h : Z→ Z , h ( x ) = x 3 x ; (d) k : R→ R , k ( x ) = e x ; (e) l : [ π/ 2 ,π/ 2]→ R , l ( x ) = sin x ; (f) j : [0 ,π ]→ R , j ( x ) = sin x . Hint : (a) f : Z→ Z , f ( x ) = 2 x + 1 is onetoone, and its range is the set of odd integers; (b) g : Q→ Q , g ( x ) = 2 x + 1 is onetoone, and its range is Q ; (c) h : Z→ Z , h ( x ) = x 3 x is not onetoone, because h (0) = h (1) = h ( 1) = 0. Its range is { n ∈ Z : n is the product of three consecutive integers } ; (d) k : R→ R , k ( x ) = e x is onetoone, and its range is R + , the set of positive real numbers; (e) l : [ π/ 2 ,π/ 2]→ R , l ( x ) = sin x is onetoone, and its range is [ 1 , 1]; (f) j : [0 ,π ]→ R , j ( x ) = sin x is not onetoone, because j (0) = j ( π ) = 0 (you may find some other witnesses), and its range is [0 , 1]. 1 3. For each of the following functions g : R→ R , determine whether the function is onetoone and whether it is onto. If the function is not onto, determine the range g ( R ). (a) g ( x ) = x + 7; (b) g ( x ) = 2 x 3; (c) g ( x ) = x + 5; (d) g ( x ) = x 2 ; (e) g ( x ) = x 2 + x ; (f) g ( x ) = x 3 . Hint : Similar to question 2....
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This note was uploaded on 01/23/2011 for the course MAS 111 taught by Professor Drchansongheng during the Spring '10 term at Nanyang Technological University.
 Spring '10
 DrChanSongHeng

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