Resposta:
. g(f(-2)) = 29 e f(g(- 2)) = 0
Explicação passo a passo:
.
. f(x) = x² - x - 20 e g(x) = 1 - 2x
.
TEMOS: f(- 2) = (- 2)² - (- 2) - 20 g(f(-2) = g(- 14)
. = 4 + 2 - 20 = 1 - 2 . (- 14)
. = 6 - 20 = 1 + 28
. = - 14 = 29
.
. g(- 2) = 1 - 2 . (- 2) f(g(-2)) = f(5)
. = 1 + 4 = 5² - 5 - 20
. = 5 = 25 - 5 - 20
. = 20 - 20
. = 0
.
(Espero ter colaborado)
✅ Após resolver os cálculos, concluímos que os valores numéricos das referidas composição de funções são, respetivamente:
[tex]\Large\displaystyle\text{$\begin{gathered}\boxed{\boxed{\:\:\:\bf g(f(-2)) = 29\:\:\:e\:\:\:f(g(-2)) = 0\:\:\:}}\end{gathered}$}[/tex]
Sejam as funções polinomiais:
[tex]\Large\begin{cases} f(x) = x^{2} - x - 20\\g(x) = 1 - 2x\\ g(f(-2)) = \:?\\f(g(-2)) = \:?\end{cases}[/tex]
Organizando as equações, temos:
[tex]\Large\begin{cases} f(x) = x^{2} - x - 20\\g(x) = -2x + 1\\g(f(-2)) = \:?\\f(g(-2)) = \:?\end{cases}[/tex]
Então, temos:
[tex]\Large\displaystyle\text{$\begin{gathered} g(f(-2)) = -2\left[f(-2)\right] + 1\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = -2\left[(-2)^{2} - (-2) - 20\right] + 1\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = -2\left[4 + 2 - 20\right] + 1\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = -2\left[-14\right] + 1\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = 28 + 1\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = 29\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} \therefore\:\:\:g(f(-2)) = 29\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} f(g(-2)) = (g(-2))^{2} - (g(-2)) - 20\end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered} = (-2\cdot(-2) + 1)^{2} - (-2\cdot(-2) + 1) - 20\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = (4 + 1)^{2} - (4 + 1) - 20\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = 5^{2} - 5 - 20\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = 25 - 5 - 20\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = 0\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} \therefore\:\:\:f(g(-2)) = 0\end{gathered}$}[/tex]
[tex]\LARGE\displaystyle\text{$\begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered}$}[/tex]
Saiba mais:
[tex]\Large\displaystyle\text{$\begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Observe \:o\:Gr\acute{a}fico!!\:\:\:}}}\end{gathered}$}[/tex]
