find the inverse and state its domain and range fx2sinpixpi

Solution

f(x)=2-sin(pi*x-pi)

domain(1/2, 3/2)

x =1/2 ,f(x)=2-sin((pi/2)-pi) =3

x =3/2 ,f(x)=2-sin((3pi/2)-pi) =1

range(1,3)

f(x)=y

x=f^-1(y)

2-sin(pi*x-pi) =y

sin(pi*x-pi) =2-y

(pi*x-pi) =sin^-1(2-y)

(x-1) =(1/pi)sin^-1(2-y)

x=1+ (1/pi)sin^-1(2-y)

f^-1(y)=1+ (1/pi)sin^-1(2-y)

f^-1(x)=1+ (1/pi)sin^-1(2-x)

domain of f^-1(x)= range of f(x)

range of f^-1(x)= domain of f(x)

domain=(1,3)

range =(1/2, 3/2)