Find the domain of the inverse function of

Question

f(x)=cos^-1(2x/3)+5 for -3/2≤x≤3/2

Answer 1

If the function f(x) could be equaled to "y" such that y=f(x) then y=cos^-1(2x/3)+5 taking 5 subtracting you have

 

y-5=cos^-1(2x/3) and now you can take inverse function on both sides and you'll have:

 

cos(y-5) = cos [cos^-1(2x/3)]  

cos (y-5) = 2x/3

3Cos(y-5)/2 = x

 

Switching x and y we have:

 

Y = 3Cos(x-5)/2 ; you can equal x-5 to Z so Z=x-5 and then you'll have Y = 3Cos(Z)/2

 

The domain for the Cos(Z) solution is that Z ∈ R which makes the domain for the original function x ∈ R