Science:Math Exam Resources/Courses/MATH220/December 2010/Question 04 (a)
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Question 04 (a) 

Let and be functions so that is an injection. Prove that must be injective. 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Recall that a function is injective if and only if for all values , we have that implies that . Use this definition for and then show that satisfies this definition. 
Hint 2 

Try a direct proof, that is, let be such that . How can you use the information about to show that ? 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The function is given to be injective, meaning that for , we have that if then necessarily . To show that is an injective function, we need to show that for any if then . Start by choosing any such that . Then, applying to both sides of shows that . This function is injective and thus . This shows that is injective completing the proof. 