Question

if f (x)=3x+5 and fog(x+2)=12x+17,find the value of x such that,gof(x)=88​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that

$\green{\rm :\longmapsto\:f(x) = 3x + 5}$

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,

$\purple{\rm :\longmapsto\:gof(x) = 88}$

$\purple{\rm :\longmapsto\:g(f(x)) = 88}$

$\purple{\rm :\longmapsto\:4f(x) - 4= 88}$

$\purple{\rm :\longmapsto\:4f(x)= 88 + 4}$

$\purple{\rm :\longmapsto\:4f(x)= 92}$

$\purple{\rm :\longmapsto\:f(x)= 23}$

$\purple{\rm :\longmapsto\:3x + 5= 23}$

$\purple{\rm :\longmapsto\:3x= 23 - 5}$

$\purple{\rm :\longmapsto\:3x= 18}$

$\purple{\bf :\longmapsto\:x= 6}$