Science:Math Exam Resources/Courses/MATH220/December 2011/Question 01 (e)
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Question 01 (e) 

Let A, B be nonempty sets, and let ƒ: A → B be a function. When does ƒ have an inverse function ƒ^{1}? Define ƒ^{1}. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
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Hint 

Why doesn't the function have an inverse? What about the function 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. A function ƒ has an inverse if and only if it is bijective. That is, it has an inverse if and only if it is
In such a case, for every y ∈ B, there is a unique x ∈ A such that ƒ(x) = y. Using this fact, we define ƒ^{1} by the rule where x is the unique element of the set A such that ƒ(x) = y. From the injectivity and surjectivity, this is well defined. Moreover, and and so this really is the inverse of ƒ. 