Assignment 2 - (c Find a formula for the inverse function...

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MATH 137 Assignment 2, Part 2 Due: 11 am, Friday, October 1 Your assignment consists of two separate parts. Part 1 is available online at http://mapleta.uwaterloo.ca and is due at 4 pm on Thursday September 30. Part 2 consists of the problems below for you to hand in. Place part 2 of your assignment in the correct drop box outside MC 4066, corresponding to the class section in which you are registered. Hand in your solutions to the following 5 problems. You must follow the same set of instructions as in Assignment 1. 1. Let f ( x ) = ax + b cx + d , where a,b,c, and d are constants. (a) When a = b = c = 1 and d = 3 , find a formula for the inverse function f - 1 ( x ) . (b) Assuming that f is one-to-one, find a formula for f - 1 ( x ) in general. (c) What condition on a,b,c, and d guarantees that f is one-to-one? (Hint: Write f ( x 1 ) = f ( x 2 ) , cross-multiply, and then try to conclude that x 1 = x 2 . We must avoid dividing by 0.) 2. Let f ( x ) = ln( x + x 2 + 1 ) . (a) Find the domain of f . (b) Prove that f is an odd function. (Hint: It is easier to check that f ( x ) + f ( - x ) = 0
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Unformatted text preview: .) (c) Find a formula for the inverse function of f . 3. Simplify the following expressions. (a) tan(arcsin x ) (b) cos(2 arctan x ) (Hint: Note that arcsin x = sin-1 x , and arctan x = tan-1 x . See Example 13 on p. 69.) 1 4. State the domains of the following functions and sketch their graphs. (a) f ( x ) = sin(arcsin x ) (b) g ( x ) = arcsin(sin x ) (Hint: For g ( x ) , note that it has period 2 . If you can draw its graph on the interval [-, ] , then you can just replicate the rest of the graph from the piece you have.) 5. Let f be a one-to-one function. For each question below, if your answer is yes, give a short proof, and if your answer is no, give an example where the statement fails. (a) Is f always either increasing or decreasing? (b) If f is increasing, is f-1 also increasing? (c) If f is decreasing, is f-1 also decreasing? (d) If f is odd, is f-1 also odd? 2...
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Assignment 2 - (c Find a formula for the inverse function...

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