Question

if f(x) =x²+1 , g(x)=3x and fog(x) = gof(x) . find the value x​

Answer 1

${\orange{ \boxed{ \boxed{ \begin{array}{cc} \bf \to \: given : \\ \\ \rm \: f(x) = {x}^{2} + 1 \\ \\ \rm \: g(x) = 3x \\ \\ \rm \: fog(x) = gof(x) \\ \\ \red{ \sf \: we \: have \: to \: find \: : } \\ \\ \rm \: value \: of \: \: x \end{array}}}}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{ \underline{ \sf \: solution \: : }}$

At first we have to find, fog(x)

${\green{ \boxed{ \boxed{ \begin{array}{cc} \rm \: fog(x) \\ \\ \rm = f(g(x)) \\ \\ \rm = {(3x)}^{2} + 1 \\ \\ \rm = 9 {x}^{2} + 1 \end{array}}}}}$

Then, we have to find, gof(x)

${\pink{ \boxed{ \boxed{ \begin{array}{cc} \rm \: gof(x) \\ \\ \rm = g(f(x)) \\ \\ \rm = 3( {x}^{2} + 1) \\ \\ \rm = 3 {x}^{2} + 3 \end{array}}}}}$

According to the question,

${\blue{ \boxed{ \boxed{ \begin{array}{cc} \rm \: fog(x) = gof(x) \\ \\ \rm \implies \: 9 {x}^{2} + 1 = 3 {x}^{2} + 3 \\ \\ \rm \implies \: 9 {x}^{2} - 3 {x}^{2} + 1 - 3 = 0 \\ \\ \rm \implies \: 6 {x}^{2} - 2 = 0 \\ \\ \rm \implies \: 2(3 {x}^{2} - 1) = 0 \\ \\ \rm \implies \: 3 {x}^{2} - 1 = 0 \\ \\ \rm \implies \: 3 {x}^{2} = 1 \\ \\ \rm \implies \: {x}^{2} = \frac{1}{3} \\ \\ \rm \implies \: x = \pm \frac{1}{ \sqrt{3} } \end{array}}}}}$