Question

Find fog gof when f(x)=2x+1 and g(x) =x^2-2

Answer 1

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

Given functions are

$\rm \:f(x) = 2x + 1 \: \:$

and

$\rm \:g(x) = {x}^{2} - 2$

Now, Consider

${\sf \:fog(x)}$

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

$\rm \: = \:f( {x}^{2} - 2)$

$\rm \: = \:2( {x}^{2} - 2) + 1$

$\rm \: = \:2{x}^{2} -4 + 1$

$\rm \: = \:2{x}^{2} -3$

${\bf\implies \:\boxed{ \sf{ \: fog(x) = {2x}^{2} - 3 \: \: }}}$

Now, Consider

${\rm\:gof(x) \: }$

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

$\rm \: = \:g(2x + 1)$

$\rm \: = \: {(2x + 1)}^{2} - 2$

$\rm \: = \: {4x}^{2} + 1 + 4x - 2$

$\rm \: = \: {4x}^{2} + 4x - 1$

${\bf\implies \:\boxed{ \sf{ \: gof(x) = {4x}^{2} + 4x - 1 \: \: }}}$

Therefore,

${\begin{gathered}\begin{gathered}\rm :\longmapsto\:\bf\:\begin{cases} &\sf{fog(x) = {2x}^{2} - 3 } \\ \\ &\sf{gof(x) = {4x}^{2} + 4x - 1} \end{cases}\end{gathered}\end{gathered}}$