How do I say that this function assumes this value?

Abhishekdas

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What I've tried-
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But, how do I prove that this function does assume the value 9. The highest power of x is 5 and I do not know how to solve such equations. Should I even need to prove that the function assumes the value 9?
 
Is there x where the value is less than 9? How about greater than 9? If they exist, can this function 'skip' y=9? See the intermediate value theorem.
 
… Should I even need to prove that the function assumes the value 9?
The need may be a function of your instructor.

All polynomials are continuous over their domain (the set of Real numbers). This has already been proven, in a number of ways. If you're allowed to reference such theorems, then I would say that you don't need to provide a proof of continuity. Your function is a 5th-degree polynomial, so it's range is (-∞,+∞).

If you're still not sure, chat up your instructor.

If you'd like to read over some different approaches to proving polynomial continuity, you can google for them. A decent working knowledge of precalculus (limits, sequences, and, of course, proofs in general -- like induction) will assist in comprehending most of the approaches. :cool:
 
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… See the intermediate value theorem …
That's a good suggestion, but it requires knowing that the function is continuous over its domain to begin with. If continuity of polynomials has already been covered (proved or given), we're off and running. :cool:
 
The need may be a function of your instructor.

All polynomials are continuous over their domain (the set of Real numbers). This has already been proven, in a number of ways. If you're allowed to reference such theorems, then I would say that you don't need to provide a proof of continuity. Your function is a 5th-degree polynomial, so it's range is (-∞,+∞).

If you're still not sure, chat up your instructor.

If you'd like to read over some different approaches to proving polynomial continuity, you can google for them. A decent working knowledge of precalculus (limits, sequences, and, of course, proofs in general -- like induction) will assist in comprehending most of the approaches. :cool:
Oh ok. I got it. Thanks a lot!
 
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