Science:Math Exam Resources/Courses/MATH220/April 2011/Question 09 (c)
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Question 09 (c) 

Let be functions so that g ○f is a surjective function. Prove or disprove that f ○g must be surjective. 
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 

Working on part (a) and part (b) definitively helps. In your example in part (b) f was not surjective. Can possibly be surjective then? 
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 

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Please rate my easiness! It's quick and helps everyone guide their studies. The example that we constructed in part (b) showed a case where the function f is not surjective and so it cannot be possible for to be surjective either. For example, in that case, no negative number is in the image of the function f and so no negative number can be in the image of either. 