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Ciara Mcdaniel 2022-07-02 Answered
Let
f ( x ) = x 6 1 3 x 1
Prove that the range of f is R .( Hint: use the Intermediate Value Theorem.)

I thought IVT was meant to show that the function has a root? Please help, I don't know how I can use IVT to prove the range.
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Answers (2)

Dobermann82
Answered 2022-07-03 Author has 15 answers
You have lim x f ( x ) = and lim x 1 3 f ( x ) = .

Moreover f is continuous on the interval ( , 1 3 ). Therefore by the IVT, the image of ( , 1 3 ) under f is equal to R . A fortiori, the image of f is equal to R .
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Wade Bullock
Answered 2022-07-04 Author has 5 answers
Hint: If y R , then asserting that y belongs to the range of f is the same thing as asserting that the equation f ( x ) y = 0 has a root.
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