a ft et31 find domain and image of f 1 b gt 2t 1 t 3 fin

a) f(t) = e^t^3+1 find domain and image of f^ -1

b) g(t) = 2t + 1/ (t + 3) find domain and image of g^ -1

Solution

a) f(t) = e^{t^3+1}

let f(t)=y =>t =f^-1(y)

e^{t^3+1}=y

t³+1=ln(y)

t³=(ln(y) -1)

t=(ln(y) -1)^1/3

f^-1(y)=(ln(y) -1)^1/3

f^-1(t)=(ln(t) -1)^1/3

f^-1(t) exists when

ln(t) -1>0

ln(t) >1

t>e

domain =(e,infinity)

image =(0,infinity)

b)g(t) = (2t + 1)/ (t + 3)

let g(t)=y

t =g^-1(y)

(2t + 1)/ (t + 3) =y

(2t + 1)=y(t + 3)

2t +1 =yt +3y

2t -yt =3y -1

t (2-y) =(3y -1)

t =(3y -1)/(2-y)

g^-1(y) =(3y -1)/(2-y)

g^-1(t) =(3t -1)/(2-t)

domain(-infinity,2)U(2,infinity)

range =domain of g(t)=(-infinity,-3)U(-3,infinity)