I found the inverse of $$f(x)=-2cos(3x)$$
$$f^{-1}(x)=\frac 13 cos^{-1}(-\frac x 2)$$
how do I find the domain? Is it all real numbers?

Question

I found the inverse of $$f(x)=-2cos(3x)$$
$$f^{-1}(x)=\frac 13 cos^{-1}(-\frac x 2)$$
how do I find the domain? Is it all real numbers?

anonymous · Answer

@satellite73

anonymous · Answer

@Callisto

callisto · Answer

Domain of... f(x) or f^(-1) (x)?

anonymous · Answer

of $$f'(x)$$

anonymous · Answer

i Mean $$f^{-1}$$

callisto · Answer

Try $f^{-1} (100)$, what would you get?

anonymous · Answer

it's an imaginary number according wolfram

callisto · Answer

I suppose the function should give real output if you input a real number.
Okay, when you put x= 100 into f^(-1) (x), it doesn't give you real output. So, it's not in the domain of y.
Consider cosine function, range of cosine function is
-1 ≤ cos x ≤ 1.
Consider the inverse of cosine function, domain of the inverse is the range of the cosine function, so the domain of cos^(-1)x is [-1, 1].
Now, you have cos^(-1) (x/2) in your inverse function, hmm.. how would you get the domain of the inverse function?

anonymous · Answer

Let's see if I understand this...
$$f^{-1}(f(x))=sin^{-1}(sinx)=x \;\;\;\;where \;\;\;\;-\frac{\pi}{2}\le x\le\frac{\pi}{2}$$
$$f(f^{-1}(x))=sin(sin^{-1}x)=x \;\;\;\;where\;\;\;\;-1\le x\le1$$

anonymous · Answer

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anonymous · Answer

I"m trying to picture this. What's bothering me is that fact the domain and range  of one function change to the range and domain of the next function....I'm trying to keep them in order somehow

phi · Answer

the cosine gives numbers between -1 and 1
there is no real number x, where cos(x)= 2 (for example)

so x = acos(2)  has no real solution.  You would want to restrict the domain to -1 to +1

for the case of acos(x/2)     -1 ≤ x/2 ≤ 1  or    -2 ≤ x ≤ 2

anonymous · Answer

but the restriction of the  inverse of cosine is $$0\le x \le \pi$$
but since it's x/2 do I multiply or divided that by two? or am i totally on the wrong path...

phi · Answer

you mean the restriction on the cosine is 0 to pi

the restriction on the domain of the inverse cos(x)  is -1 ≤ x ≤ 1

phi · Answer

the restriction on the domain of the inverse cos(x)  is -1 ≤ x ≤ 1

if you are given  acos(x/2)   then rename the variable  to y= x/2
you know the domain is restricted to
-1 ≤ y ≤ 1

but with y= x/2
-1 ≤ x/2 ≤ 1

solve for x, (2 separate relations), to get
-2 ≤ x ≤ 2
that is the domain in terms of x for acos(x/2)

anonymous · Answer

I find gauss's law easier to understand than this for whatever reason. I'm very visual and I can't visualize this....I would much rather draw unit circles and sine graphs, and modify that to represent the current problem that we're working on.

anonymous · Answer

I'll look at it again later...too tired to comprehend.