Question

Se f e g são funções de R em R dadas pelas leis f(x)=x³ e g(x)=x+1, obtenha as leis que definem as compostas: g o f, f o g, f o f e g o g.

Answer 1

✅ Sejam as funções polinomiais:

               $\Large\begin{cases} f(x) = x^{3}\\g(x) = x + 1\end{cases}$      

Calculando as composições das respetivas funções, temos:

• g o f:

           $\Large\displaystyle\begin{gathered} g o f = g(f(x))\end{gathered}$

                     $\Large\displaystyle\begin{gathered} = f(x) + 1\end{gathered}$

                     $\Large\displaystyle\begin{gathered} = x^{3} + 1\end{gathered}$

• f o g:

            $\Large\displaystyle\begin{gathered} f o g = f(g(x))\end{gathered}$

                      $\Large\displaystyle\begin{gathered} = (x + 1)^{3}\end{gathered}$

                      $\Large\displaystyle\begin{gathered} = x^{3} + 3x^{2} + 3x + 1\end{gathered}$

• f o f:

            $\Large\displaystyle\begin{gathered} f o f = f(f(x))\end{gathered}$

                      $\Large\displaystyle\begin{gathered} = (f(x))^{3}\end{gathered}$

                      $\Large\displaystyle\begin{gathered} = (x^{3})^{3}\end{gathered}$

                       $\Large\displaystyle\begin{gathered} = x^{9}\end{gathered}$

• g o g:

             $\Large\displaystyle\begin{gathered} g o g = g(g(x))\end{gathered}$

                      $\Large\displaystyle\begin{gathered} = g(x) + 1\end{gathered}$

                      $\Large\displaystyle\begin{gathered} = x + 1 + 1\end{gathered}$

                      $\Large\displaystyle\begin{gathered} = x + 2\end{gathered}$

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

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$\Large\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Observe \:o\:Gr\acute{a}fico!!\:\:\:}}}\end{gathered} }$