Winter 2008 - Cioaba's Class - Exam 2 (Version 2)

Winter 2008 - Cioaba's Class - Exam 2 (Version 2) - Name...

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Name: PID: Midterm 2, Math 109 - Winter 2008 Duration: 50 minutes Please close your books, turn off your calculators and phones. To get full credit you should explain your answers. 1. Let f : R R f ( x ) = 5 x + 3 and g : R R , g ( x ) = x 4 be two functions. a.(2 points) Find f g and g f . Proof. We have ( f g )( x ) = f ( g ( x )) = f ( x 4 ) = 5 x 4 + 3 and ( g f )( x ) = g ( f ( x )) = g (5 x + 3) = (5 x + 3) 4 b.(3 points) Show that f is a bijective function and find its inverse f - 1 . Show that g is not a bijective function. Proof. f is injective because if f ( x 1 ) = f ( x 2 ), this means that 5 x 1 + 3 = 5 x 2 + 3. It follows that 5 x 1 = 5 x 2 which implies that x 1 = x 2 . f is surjective because for any y R , if f ( x ) = y , then 5 x + 3 = y which implies that x = y - 3 5 . Thus, f is bijective and its inverse is f - 1 ( y ) = y - 3 5 . Because g (1) = g ( 1) = 1 and 1 negationslash = 1, it follows that g is not injective. Thus, g is not bijective.
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2.a.(2 points) Show that among any 6 distinct numbers from the set { 1 , 2 ,..., 10 } , there will be two whose sum is 11.
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