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inverse functions - MAC1105 Inverse Functions When a task...

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MAC1105: Inverse Functions When a task involves several steps, we may often “undo” the task by undoing the steps in reverse order. For example, in the morning we might put on socks, and then shoes. To undo this task we must first take off the shoes, and then the socks. If we drive from home to work by driving four miles north and then five miles east, we may get back home by driving five miles west and then four miles south. To undo a multi-step task, we undo the steps in reverse order. Example 1: For each of the following multi-step tasks, write down the sequence of steps that would “undo” the task. a. You walk out your front door, get in your car, start the engine, and drive to the store. b. You buy a TV stand, take it home, unpack it, put it together, and place the TV on top of it. Solutions a. You drive home from the store, turn off the car engine, get out of the car, and walk in your front door. b. You take the TV off the stand, take the stand apart, put it back in the box, take it back to the store, and return it. We can extend this idea to mathematical functions. For every mathematical operation, there is another operation that “undoes” it. For example, if we add five to a number, we can subtract five from the result to obtain our original number. If we triple a number, we can divide the result by three to obtain our original number. Example 2: Find a function g that undoes each of the following. Verify by finding formulas for g f ° and f g ° . a. f(x) x 5 = + = b. f(x) 3x = c. f(x) 3x 5 = + = + = + = + Solutions a. Function f adds 5 to the input. Define a function which subtracts 5, namely g(x) x 5 = - . In section 5.1 we saw that the operation of composition is not generally commutative. However, using f and g as defined above: ( ) ( ) ( ) ( ) ( ) ( ) (g f )(x) g f(x) g x 5 x 5 5 x (f g)(x) f g(x) f x 5 x 5 5 x = = + = + - = = = + = + - = = = = = = = - = - + = ° ° Not only are the results of the compositions the same, but the fact that they both simplify to x indicates that each operation undoes the other. That is, subtracting five undoes the addition of five, and adding five undoes the subtraction of five.
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2 b. Function f multiplies the input by 3. Define a function that divides by 3, namely x g(x) 3 = . Proceeding as before, we obtain a similar result. ( ) ( ) ( ) 3x (g f )(x) g f(x) g 3x x 3 x x (f g)(x) f g(x) f 3 x 3 3 = = = = = = ° ± ° ± ° ± ° ± = = = = ² ³ ² ³ ´ µ ´ µ ° ° c. This is a two-step function. Undoing each step in reverse order will generate the steps for g . f g 1. multiplies by 3 1. subtracts 5 2. adds 5 2. divides by 3 Using the above steps for g we can now write x 5 g(x) 3 - = . Let’s verify that these two functions undo one another. ( ) ( ) ( ) ( ) ( ) 3x 5 5 3x (g f )(x) g f(x) g 3x 5 x 3 3 x 5 x 5 (f g)(x) f g(x) f 3 5 x 5 5 x 3 3 + - - = = + = = = - - - - ° ± ° ± ° ± ° ± ± ± = = = + = - + = = ² ³ ² ³ ² ³ ² ³ ³ ³ ´ µ ´ µ µ ° ° Functions which undo one another are called inverses .
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