MATH220 December 2010
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Question 05 (b)
Prove that the following function 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!
Recall that a function is bijective if it is one to one and onto.
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We first show our function is one to one (injective).
Let be such that , that is,
Cross multiplying gives
and cancelling like terms shows that
To show our function is onto, suppose that . We want to know if there is a such that , that is,
If we isolate for our x, we see that
Notice in the last step dividing by is allowed since . Defining implies that . It remains to show that , that is, . Using a proof by contradiction we assume that , which implies
A contradiction! Hence, indeed, and the proof is complete.
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