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Math Help - Another conundrum :s If f(x)=cos(arcsinx), what is the range of f?

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    Another conundrum :s If f(x)=cos(arcsinx), what is the range of f?

    If f(x)=cos(arcsinx), what is the range of f?

    What makes this even more confusing to me, is that the multiple choice answers are in some sort of weird form.

    {x|-1 <or= x <or= 0}

    I thought that range was supposed to be in the y-axis? I have no clue what the "x|" in the beginning means.
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    Quote Originally Posted by JOhkonut View Post
    If f(x)=cos(arcsinx), what is the range of f?
    Remember the max value of \cos is 1 and smallest is -1. Thus the range is at most between -1 and 1. We can reach 1 because if we let x=0 then \arcsin 0 = 0 and \cos 0 = 1. But can we reach -1? No! Because \arcsin x always gives a number between -\pi/2 and \pi/2. And you have to remember \cos is non-negative for these values. Therefore 0 is the low point. Thus the range is \{ y | 0\leq y\leq 1\}.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by JOhkonut View Post
    If f(x)=cos(arcsinx), what is the range of f?

    What makes this even more confusing to me, is that the multiple choice answers are in some sort of weird form.

    {x|-1 <or= x <or= 0}

    I thought that range was supposed to be in the y-axis? I have no clue what the "x|" in the beginning means.
    recall that the range of arcsin(x) is \bigg[ - \frac {\pi}2, \frac {\pi}2 \bigg]. so these will be your input values. now the question is, what values does cos(x) take on when - \frac {\pi}2 \le x \le \frac {\pi}2?

    hint: look at the graph of cosine on that domain. what is the range of y-values that it hits?
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    Quote Originally Posted by Jhevon View Post
    recall that the range of arcsin(x) is \bigg[ - \frac {\pi}2, \frac {\pi}2 \bigg]. so these will be your input values. now the question is, what values does cos(x) take on when - \frac {\pi}2 \le x \le \frac {\pi}2?

    hint: look at the graph of cosine on that domain. what is the range of y-values that it hits?
    Ok, I got that far with you now, but is there any way to explain why that is the domain of arcsin? Or is it just something that I have to know?

    And are the multiple choices typos or something, because I still don't understand the whole {x| ... thing.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by JOhkonut View Post
    Ok, I got that far with you now, but is there any way to explain why that is the domain of arcsin? Or is it just something that I have to know?

    And are the multiple choices typos or something, because I still don't understand the whole {x| ... thing.
    yes, the range is something you should know. you should see your text or try google to see how it is derived.

    as for the notation: \{ x \mid -1 \le x \le 0 \} means "the set of all x such that (that's what the | means) x is between -1 and 0 inclusive"
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