If f (x)=3x+5 and fog(x+2)=12x+17,find the value of x such that,gof(x)

1.	If f (x)=3x+5 and fog(x+2)=12x+17,find the value of x such that,gof(x)=88
Answer» $\large\underline{\sf{Solution-}}$ GIVEN that $\green{\rm :\longmapsto\:f(<klux>X</klux>) = 3x + 5}$ So, $\green{\bf :\longmapsto\:f(g(x)) = 3g(x) + 5 - - - (1)}$ Now, Further given that $\red{\rm :\longmapsto\:fog(x + 2) = 12x + 17}$ Put x + 2 = y, so that x = y - 2 So, $\red{\rm :\longmapsto\:fog(y) = 12(y - 2)+ 17}$ $\red{\rm :\longmapsto\:fog(y) = 12y - 24+ 17}$ $\red{\rm :\longmapsto\:fog(y) = 12y -7}$ $\red{\rm :\longmapsto\:f(g(y)) = 12y -7}$ $\red{\bf :\longmapsto\:f(g(x)) = 12x - 7 - - - (2)}$ Now, EQUATING equation (2) and (1), we GET $\blue{\rm :\longmapsto\:3g(x) + 5 = 12x - 7}$ $\blue{\rm :\longmapsto\:3g(x)= 12x - 7 - 5}$ $\blue{\rm :\longmapsto\:3g(x)= 12x - 12}$ $\blue{\bf :\longmapsto\:g(x)= 4x - 4 - - - (3)}$ So, it means $\bf\implies \:g(f(x) = 4f(x) - 4$ Further, given that, $\purple{\rm :\longmapsto\:gof(x) = 88}$ $\purple{\rm :\longmapsto\:g(f(x)) = 88}$ $\purple{\rm :\longmapsto\:4f(x) - 4= 88}$ $\purple{\rm :\longmapsto\:4f(x)= 88 + 4}$ $\purple{\rm :\longmapsto\:4f(x)= 92}$ $\purple{\rm :\longmapsto\:f(x)= 23}$ $\purple{\rm :\longmapsto\:3x + 5= 23}$ $\purple{\rm :\longmapsto\:3x= 23 - 5}$ $\purple{\rm :\longmapsto\:3x= 18}$ $\purple{\bf :\longmapsto\:x= 6}$