Question

let f and g be the two functions from R to R defined by f(x)=3x-4 and g(x)=x³+3.find g o f and f o g​

Answer 1

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

Given that

$\rm :\longmapsto\:f(x) = 3x - 4$

and

$\rm :\longmapsto\:g(x) = {x}^{3} + 3$

Now, Consider

$\rm :\longmapsto\:g \: o \: f$

$\rm \: = \:g \: o \: f \: (x)$

$\rm \: = \:g\bigg(f(x)\bigg)$

$\rm \: = \:g\bigg(3x - 4\bigg)$

$\rm \: = \: {(3x - 4)}^{3} + 3$

$\rm \: = \: {(3x)}^{3} - 3 {(3x)}^{2}(4) + 3(3x)( {4)}^{2} - {4}^{3} + 3$

$\rm \: = \: {27x}^{3} - 108 {x}^{2} + 144x - 64 + 3$

$\rm \: = \: {27x}^{3} - 108 {x}^{2} + 144x - 61$

$\rm :\longmapsto\:\boxed{ \bf{ \: g \: o \: f(x) = \: {27x}^{3} - 108 {x}^{2} + 144x - 61}}$

Now, Consider

$\rm :\longmapsto\:f \: o \: g$

$\rm \: = \:\:f \: o \: g \: (x)$

$\rm \: = \:f\bigg(g(x)\bigg)$

$\rm \: = \:f\bigg( {x}^{3} + 3\bigg)$

$\rm \: = \:3( {x}^{3} + 3) - 4$

$\rm \: = \:3{x}^{3} + 9 - 4$

$\rm \: = \:3{x}^{3} + 5$

$\rm :\longmapsto\:\boxed{ \bf{ \: f \: o \: g(x) = \:3{x}^{3} + 5}}$

Additional Information :-

$\boxed{ \bf{ \:f o g (x) = x \: \bf\implies \:g(x) = {f}^{ - 1}(x)}}$

$\boxed{ \bf{ \:g o f (x) = x \: \bf\implies \:f(x) = {g}^{ - 1}(x)}}$