Science:Math Exam Resources/Courses/MATH110/April 2018/Question 02 (b)
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Question 02 (b) 

For what value of the constant is the function continuous everywhere? 
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 

When do we have ? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The domain of the function is , because any polynomials are defined on the real line and the rational function is defined on . However, the domain of the function is because in order to compute , we have to use the branch as this one is defined for . (so ) On the domain, observe that for any value of , the function will be continuous for all and for all , so the only point to check is . The function will be continuous at if and only if which by definition is equivalent to We have and so will be continuous at if and only if . After solving for , we conclude that is continuous everywhere if and only if .
Answer: The function is continuous everywhere if and only if . 