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The Intermediate Value Theorem has been proved already: a continuous function on an interval $[a,b]$ attains all values between $f(a)$ and $f(b)$. Now I have this problem:

Verify the Intermediate Value Theorem if $f(x) = \sqrt{x+1}$ in the interval is $[8,35]$.

I know that the given function is continuous throughout that interval. But, mathematically, I do not know how to verify the theorem. What should be done here?

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    Perhaps more to the point, do you understand what is being asked for in this question, or is that where you’re stuck?2012-11-18
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    I know it is continuous throughout that interval. But, mathematically I do not know how to verify it.2012-11-18
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    You verify it as follows: First, note that $f(8) = 3$ and $f(35) = 6$. Then let $y$ be any real number in $[3,6]$, and show that there exists $x \in [8,35]$ such that $f(x)=y$. In shis case, $x$ can be expressed as a simple function of $y$.2012-11-18

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