MAS111_09_assignment_13_1_hints

MAS111_09_assignment_13_1_hints - MAS 111 FOUNDATION OF...

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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 one-to-one 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 one-to-one, and its range is the set of odd integers; (b) g : Q-→ Q , g ( x ) = 2 x + 1 is one-to-one, and its range is Q ; (c) h : Z-→ Z , h ( x ) = x 3- x is not one-to-one, 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 one-to-one, and its range is R + , the set of positive real numbers; (e) l : [- π/ 2 ,π/ 2]-→ R , l ( x ) = sin x is one-to-one, and its range is [- 1 , 1]; (f) j : [0 ,π ]-→ R , j ( x ) = sin x is not one-to-one, 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 one-to-one 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.

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MAS111_09_assignment_13_1_hints - MAS 111 FOUNDATION OF...

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