Find the inverse of the function and state its domain fx 3

Find the inverse of the function and state its domain. f(x) = 3 sin^-1(x/7) + 2 for -7 lessthanorequalto x lessthanorequalto 7 f^-1(x) = 3 sin (x/7) + 2 for [- pi/7, pi/7] f^-1(x) = 7 sin (x - 2/3) for [2 - pi/2, 2 + pi/2] Solve: From a balloon 1197 feet high, the angle of depression to the ranger headquarters is 50 degree 21\' How far is the headquarters from a point on the ground directly below the balloon (to the nearest foot)? Solve: A police helicopter is monitoring the speed of two cars on a straight road. The helicopter is at an altitude of 4500 feet directly above the road. At one instant, the angle of elevation from the first car to the helicopter is 20 degree, and the angle of elevation from the second car to the helicopter is 15 degree. How far apart are the two cars to the nearest foot? 4130 ft 4327 ft 431 ft 4230 ft

Solution

f(x) = 3sin^-1(x/7) +2 for -7 <= x <= 7

at x = -7 ; f(x) = 3sin^-1(-7/7) +2 = 3(-pi/2) +2 = 2 -3pi/2

at x= 7 ; f(x) = 3sin^-1(7/7) +2 = 3(pi/2) +2 = 2 + 3pi/2

This would define the domain of inverse function: [ 2 -3pi/2 , 2+3pi/2 ]

Let y = 3sin^-1(x/7) +2

Plug x = y and y = x

So, x = 3sin^-1(y/7) +2

(x -2) = 3sin^-1(y/7)

sin^-1(y/7) = ( x - 2)/3

y/7 = sin[(x-2)/3]

y = 7sin[(x-2)/3]    for [ 2 -3pi/2 , 2+3pi/2 ]

f^-1(x) = 7sin[(x-2)/3]    for [ 2 -3pi/2 , 2+3pi/2 ]

Option d