Science:Math Exam Resources/Courses/MATH220/April 2011/Question 05
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Question 05 

Prove that the function given by is bijective. 
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? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Remember that bijective is the same as being both injective (onetoone) and surjective (onto). 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. We will first show that this function is injective. Suppose that for some . First, we note that if then . So we can suppose that . Then we would have which is true if and only if , and so the function is injective. To show that the function is surjective, we need to "invert" the function. That is, we want to show that for each that we can find some so that . Expanding this yields
Hence we can choose . In such a case, we have as desired. (Note that this final justification is included only as a sanity check and is not needed in a solution to obtain full marks). 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. Recall that a function is bijective if and only if it has an inverse function, that is, a function such that both
and that
I claim that for
the function
is an inverse function for . To prove this, we proceed directly. Notice that
and that
and this completes the proof. To see where came from, check out solution 1. 