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$$\begin{align*} f(x) &= \frac{2}{x+1}, \\ g(x) &= \cos x, \\ h(x) &= \sqrt{x+3} \end{align*} $$ Find the range of $f(g(h(x)))$.

Please explain the problem.

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Just to make sure.. You should review the concepts of domain and range of a function. –  user2468 May 14 '12 at 19:17
    
@SauravTomar until OP indicates so, we can not preemptively put the (homework) tag on posts. –  user2468 May 14 '12 at 19:20
    
But this OP is a relatively new one, so I thought it was appropriate to tag his question as he might not be very familiar with the system here on MSE. –  Tomarinator May 14 '12 at 19:26
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@SauravTomar I understand. I have wondered about the same thing before, and I was told not do it :) –  user2468 May 14 '12 at 19:28
    
Dear Sk D Champ, is this a homework question? If so, please add the tag (homework). –  user2468 May 14 '12 at 19:29
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1 Answer

You want to all the possible "outputs" of the composition of the three functions. First note that the range of the inner most function $h$ is all non-negative numbers. Hence the "input" for the function $g$ is all non-negative real numbers. But for those you in $g(x) = \cos(x)$ get all real numbers between $-1$ and $1$ (both included). Note now that the domain of $f$ is all the real numbers that are not equal to $-1$, hence the possible "inputs" of $f$ is the interval $(-1, 1]$.

So now you just need to determine the possible values of $f$ when $x$ is in $(-1, 1]$. Hint: For this note that $f$ is a decreasing function.

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