Question

Seja as funções reais definidas pelas leis a seguir:
f(x) =
(2x, sex S-2
(x2-3, sex > -2
eg(x) = -2x + 1
Assim, o valor de fog-1(5) é?
Resposta:

Answer 1

$\large\boxed{\begin{array}{l}\rm Seja~as~func_{\!\!,}\tilde oes~reais~d~\!\!efinidas~pelas~leis~a~seguir:\\\boxed{\rm f(x)=\begin{cases}\rm 2x,~~se~~~~~~~~x\leqslant-2\\\rm x^2-3,~~se~~~x>-2\end{cases}~~e~~~~~~~g(x)= -2x+1}\\\\\rm Assim,o~valor~de~f\circ g^{-1}(5)~\acute e:\end{array}}$

$\large\boxed{\begin{array}{l}\boldsymbol{Resposta:}~\sf1\\\\\boldsymbol{Explicac_{\!\!,}\tilde ao~passo~a~passo:}\\\sf Como~estamos~interessados~em~saber\\\sf o~valor~da~composta~de~f~com~a~inversa~de~g\\\sf no~ponto~5,a~sentenc_{\!\!,}a~de~f~que~vai~possibilitar\\\sf tal~composta~\acute e~a~segunda~pois~5>-2.\\\sf DA\acute I\\\sf g(x)=-2x+1\\\sf 2x=1-g(x)\\\sf x=\dfrac{1-g(x)}{2}\implies g^{-1}(x)=\dfrac{1-x}{2}\\\\\sf f\circ g^{-1}(x)=f[g^{-1}(x)]=\bigg(\dfrac{1-x}{2}\bigg)^2-3\end{array}}$

$\large\boxed{\begin{array}{l}\sf f\circ g^{-1}(5)=\bigg(\dfrac{1-5}{2}\bigg)^2-3=\bigg(-\dfrac{4}{2}\bigg)^2-3\\\\\sf f\circ g^{-1}(5)=(-2)^2-3\\\sf f\circ g^{-1}(5)=4-3\\\huge\boxed{\boxed{\boxed{\boxed{\sf f\circ g^{-1}(5)=1}}}}\end{array}}$