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# MATH 107 Mathematics Given Pair of Functions Solved Quiz 5

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Name: ____________________ Date: _____________________ MATH 107 Quiz 5 3 1) For f (x) = √x + 1, g(x) = 4x2 −x find: a. (g ◦ f )(0) b. (f ◦ g)( 21 ) c. (f ◦ f )(− 2) 3 2 3 a. (g ◦ f )(x) = g (f (x)) = 4(√x + 1) −(√x + 1) 2 3 3 g (f (0)) = 4(√0 + 1) −(√0 + 1) 2 3 3 = 4(√1) −(√1) = 4(1)2 −(1) = 4 −1 = 3 b. (f ◦ g)(x) = f (g(x)) = = f (g( 21 )) = 2 √ 3 3 √( 4x2 −x) + 1 ( 4( 21 ) − 21 ) + 1 √( 4( ) − ) + 1 = √( 1 − ) + 1 3 = 1 4 3 1 2 1 2 3 = √1.5 ≈ 1.14 c. (f ◦ f )(x) = f (f (x)) = = f (f (− 2)) = = √ 3 3 3 √ 3 (√− 1) + 1 = √(− 1) + 1 3 = √0 = 0 3 √ 3 3 (√x + 1) + 1 (√− 2 + 1) + 1 2) Use the given pair of functions f (x) = 3 − x2 , g(x) = √x + 1 to find and simplify the expression (g ◦ f )(x) and state the domain of each using interval notation. (g ◦ f )(x) = g (f (x)) = √(3 − x2 ) + 1 = √4 − x2 Looking at the graph of (g ◦ f )(x) , Domain: [− 2, 2] and Range: [0, 2] 3) Show that the given function is one-to-one and find its inverse. Check your answers algebraically and graphically. f (x) = x 1 − 3x . For showing one-to-one, f(x) must pass the vertical and horizontal line test. The graph below shows that to be true. The x=1 vertical line passes through graph once and y=1 horizontal line passes through graph once. For checking one-to-one​ ​algebraically​, I f f (a) = f (b) implies that a = b then f is 1 − to − 1 . a b 1? ...
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