Inverse Functions

# Remember that an inverse function reverses the inputs

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Remember that an inverse function reverses the inputs and outputs. When we graph functions, we always represent the incoming number as x and the outgoing number as y. So to find the inverse function, switch the x and y values, and then solve for y. Example: Building and Testing an Inverse Function 1. Find the inverse function of a. Write the function as

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b. Switch the x and y variables. c. Solve for y. 5x = 2y – 3 5x + 3 = 2y . So, . 2. Test to make sure this solution fills the definition of an inverse function. a. Pick a number, and plug it into the original function. 9 f (x) 3. b. See if the inverse function reverses this process. 3 f –1 (x) 9. It worked! Were you surprised by the answer? At first glance, it seems that the numbers in the original function (the 2, 3, and 5) have been rearranged almost at random. But with more thought, the solution becomes very intuitive. The original function f (x) described the following process: double a number, then subtract 3, then divide by 5 . To reverse this process, we need to reverse each step in order: multiply by 5, then add 3, then divide by 2 . This is just what the inverse function does. Some functions have no inverse function Some functions have no inverse function. The reason is the rule of consistency. For instance, consider the function y = x2. This function takes both 3 and –3 and turns them into 9. No problem: a function is allowed to turn different inputs into the same output . However, what does that say about the inverse of this particular function? In order to fulfill the requirement of an inverse function, it would have to take 9, and turn it into both 3 and –3, which is the one and only thing that functions are not allowed to do. Hence, the inverse of this function would not be a function at all!
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