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 August 14th, 2013, 12:19 PM #1 Senior Member     Joined: Jan 2012 Posts: 123 Thanks: 2 Range of a function HI, How can we find the range of the following function: $f(x)=\sin \left \{ \ln \left ( \frac{\sqrt{4-x^{2}}}{1-x} \right ) \right \}$ However, I found its domain to be $x\epsilon \left ( -2, 1 \right )$.
 August 14th, 2013, 01:39 PM #2 Global Moderator   Joined: May 2007 Posts: 6,680 Thanks: 658 Re: Range of a function Do it in two steps. First find the range of ln(arg). If it is more than 2? in length, then the range of the sin is [-1,1].
 August 15th, 2013, 02:18 AM #3 Senior Member     Joined: Jan 2012 Posts: 123 Thanks: 2 Re: Range of a function Range of ln(arg) is of course, R. But how it is varying with the sine function...is something I am not able to arrive at.
 August 15th, 2013, 02:21 AM #4 Senior Member     Joined: Jan 2012 Posts: 123 Thanks: 2 Re: Range of a function And when $x\epsilon \left ( -2, 1 \right )$, $\sin \epsilon \left ( -1,sin1 \right )$ but then how to interpret with ln(arg)?
 August 15th, 2013, 01:23 PM #5 Global Moderator   Joined: May 2007 Posts: 6,680 Thanks: 658 Re: Range of a function When the domain of a sine function is the entire real line, the sine simply oscillates between -1 and 1. Therefore the range is [-1,1]. I don't understand what your concern is?
August 16th, 2013, 07:56 AM   #6
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Re: Range of a function

Quote:
 Originally Posted by mathman When the domain of a sine function is the entire real line, the sine simply oscillates between -1 and 1. Therefore the range is [-1,1]. I don't understand what your concern is?
Well, the domain of the sine function is R but then what the ln(arg) part playing the role..? Is it not affecting the range?

 August 16th, 2013, 08:40 AM #7 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,912 Thanks: 1110 Math Focus: Elementary mathematics and beyond Re: Range of a function You need to compute the range of $\ln$$\frac{\sqrt{4\,-\,x^2}}{1\,-\,x}$$$ on the interval (-2, 1), then consider the sine of that range of values.
August 16th, 2013, 09:31 AM   #8
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Re: Range of a function

Quote:
 Originally Posted by greg1313 You need to compute the range of $\ln$$\frac{\sqrt{4\,-\,x^2}}{1\,-\,x}$$$ on the interval (-2, 1), then consider the sine of that range of values.
Let its range be g. Then $g\epsilon \left (- \infty ,\infty \right )$. Is it correct...?

 August 17th, 2013, 06:32 AM #9 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,912 Thanks: 1110 Math Focus: Elementary mathematics and beyond Re: Range of a function Yes.
 August 17th, 2013, 08:31 AM #10 Senior Member     Joined: Jan 2012 Posts: 123 Thanks: 2 Re: Range of a function It means the answer to the original question is $y\epsilon \left ( -1,1 \right )$ but the answer give in the text book is $y\epsilon \left ( -1,\sin 1 \right )$ which should be misprinted, I believe...correct..?

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