Inverse Functions

# 3 9 3 9 if 3 goes in 9 comes out if 3 goes in 9 also

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• apiccirello
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3 9 –3 9 If 3 goes in, 9 comes out. If –3 goes in, 9 also comes out. No problem: 9 3 9 –3 But its inverse would have to turn 9 into both 3 and –3. No function can do this, so there is no inverse. In general, any function that turns multiple inputs into the same output does not have an inverse function. What does that mean in the real world? If we can convert Fahrenheit to Celsius, we must be able to convert Celsius to Fahrenheit. If we can ask “How much money did Alice make in 3 days?” we

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must surely be able to ask “How long did it take Alice to make \$500?” When would you have a function that cannot be inverted? Let’s go back to this example: Recall the example that was used earlier: “Max threw a ball. The height of the ball depends on how many seconds it has been in the air.” The two variables here are h (the height of the ball) and t (the number of seconds it has been in the air). The function h (t) enables us to answer questions such as “After 3 seconds, where is the ball?” The inverse question would be “At what time was the ball 10 feet in the air?” The problem with that question is it may well have two answers ! The ball is here... ...after this much time has elapsed 10 ft 2 seconds (*on the way up) 10 ft 5 seconds (*on the way back down) So what does that mean? Does it mean we can’t ask that question? Of course not. We can ask that question, and we can expect to mathematically find the answer, or answers—and we will do so in the quadratic chapter. However, it does mean that time is not a function of height because such a “function” would not be consistent: one question would produce multiple answers.
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• Fall '12
• JenniferSera
• Inverse Functions, Continuous function, Inverse function, Alice, Function composition

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