tut7a-large - + b . If ( g f )( x ) = 9 x 2-9 x + 3,...

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Problems to Week 8 Tutorial — MACM 101 1. Determine whether or not the following relations are functions. If a relation is a function, find its range. (a) { ( x,y ) | x,y Z , y = x 2 + 7 } , a relation from Z to Z ; (b) { ( x,y ) | x,y R , y 2 = x } , a relation from R to R . 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) f : Q Q , f ( x ) = 2 x + 1; (c) f : Z Z , f ( x ) = x 3 - x ; 3. Let f : A B where A = X Y with X Y = . If f | X and f | Y are one-to-one, does it follow that f is one-to-one? 4. For each of the following functions f : Z Z , determine whether the function is onto. If the function is not onto, determine the range of f . (a) f ( x ) = 2 x - 3; (b) f ( x ) = x 2 + x . 5. Let f,g : R R , where g ( x ) = 1 - x + x 2 and f ( x ) = ax
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Unformatted text preview: + b . If ( g f )( x ) = 9 x 2-9 x + 3, determine a,b . 6. Let g : N N be dened by g ( n ) = 2 n . If A = { 1 , 2 , 3 , 4 } and f : A N is given by f = { (1 , 2) , (2 , 3) , (3 , 5) , (4 , 7) } , nd g f . 7. If f g is onto, does it follow that g is onto? 8. For each of the following functions f : R R , determine whether f is invertible, and, if so, determine f-1 . (a) f = { ( x,y ) | 2 x + 3 y = 7 } ; (b) f = { ( x,y ) | y = x 4 + x } . 9. Show that (a) 5 n + 8 O ( n + 1); (b) n 2 6 O ( n 2 ); (c) n 2 o ( n 2 ). 1...
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