Math, asked by khanarbaz1568, 1 month ago

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

Answers

Answered by TrustedAnswerer19
7

{\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}}}}}

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