First, let's evaluate the inverse of #f(t)#, i.e. #f^(- 1)(t)#:
#Rightarrow f(t) = - frac(3)(2 t + 1)#
Let's replace #f(t)# with #s#:
#Rightarrow s = - frac(3)(2 t + 1)#
Interchanging variables:
#Rightarrow t = - frac(3)(2 s + 1)#
We need to solve this equation for #s#:
#Rightarrow - frac(t)(3) = frac(1)(2 s + 1)#
#Rightarrow (- frac(t)(3))^(- 1) = (frac(1)(2 s + 1))^(- 1)#
#Rightarrow - frac(3)(t) = 2 s + 1#
#Rightarrow 2 s = - frac(3)(t) - 1#
#Rightarrow 2 s = - frac(3)(t) - frac(t)(t)#
#Rightarrow 2 s = - frac(3 + t)(t)#
#Rightarrow s = - frac(3 + t)(2 t)#
Let's replace #s# with #f^(- 1)(t)#:
#therefore f^(- 1)(t) = - frac(t + 3)(2 t)#
Now, let's evaluate #g(f^(- 1)(t))#
#Rightarrow g(f^(- 1)(t)) = g(- frac(t + 3)(2 t))#
#Rightarrow g(f^(- 1)(t)) = frac(7)((- frac(t + 3)(2 t))^(2))#
#Rightarrow g(f^(- 1)(t)) = frac(7)(frac((t + 3)^(2))(4 t^(2)))#
#Rightarrow g(f^(- 1)(t)) = 7 cdot frac(4 t^(2))((t + 3)^(2))#
#therefore g(f^(- 1)(t)) = frac(28 t^(2))((t + 3)^(2))#; #t ne - 3#
