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

1. ## 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.

2. Originally Posted by JOhkonut
If f(x)=cos(arcsinx), what is the range of f?
Remember the max value of $\displaystyle \cos$ is $\displaystyle 1$ and smallest is $\displaystyle -1$. Thus the range is at most between $\displaystyle -1$ and $\displaystyle 1$. We can reach $\displaystyle 1$ because if we let $\displaystyle x=0$ then $\displaystyle \arcsin 0 = 0$ and $\displaystyle \cos 0 = 1$. But can we reach $\displaystyle -1$? No! Because $\displaystyle \arcsin x$ always gives a number between $\displaystyle -\pi/2$ and $\displaystyle \pi/2$. And you have to remember $\displaystyle \cos$ is non-negative for these values. Therefore $\displaystyle 0$ is the low point. Thus the range is $\displaystyle \{ y | 0\leq y\leq 1\}$.

3. Originally Posted by JOhkonut
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 $\displaystyle \bigg[ - \frac {\pi}2, \frac {\pi}2 \bigg]$. so these will be your input values. now the question is, what values does cos(x) take on when $\displaystyle - \frac {\pi}2 \le x \le \frac {\pi}2$?

hint: look at the graph of cosine on that domain. what is the range of y-values that it hits?

4. Originally Posted by Jhevon
recall that the range of arcsin(x) is $\displaystyle \bigg[ - \frac {\pi}2, \frac {\pi}2 \bigg]$. so these will be your input values. now the question is, what values does cos(x) take on when $\displaystyle - \frac {\pi}2 \le x \le \frac {\pi}2$?

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.

5. Originally Posted by JOhkonut
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: $\displaystyle \{ x \mid -1 \le x \le 0 \}$ means "the set of all x such that (that's what the | means) x is between -1 and 0 inclusive"