Question

dadas f(x)=3x+5 e g(x)=2x-3, calcule x para que se tenha.
a)f(g(x))=0
b)g(f(x))=1​

Answer 1

Determinar as compostas fog(x) e gof(x)

fog(x):

                     $\sf f\big(g(x)\big)~=~3\cdot g(x)~+~5\\\\\sf f\big(g(x)\big)~=~3\cdot (2x-3)~+~5\\\\\sf f\big(g(x)\big)~=~3\cdot 2x~+~3\cdot (-3)~+~5\\\\\sf f\big(g(x)\big)~=~6x~-~9~+~5\\\\\boxed{\sf \sf f\big(g(x)\big)~=~6x~-~4}$

• gof(x):

                     $\sf g\big(f(x)\big)~=~2\cdot f(x) ~-~3\\\\\sf g\big(f(x)\big)~=~2\cdot (3x+5) ~-~3\\\\\sf g\big(f(x)\big)~=~2\cdot 3x~+~2\cdot 5 ~-~3\\\\\sf g\big(f(x)\big)~=~6x ~+~10~-~3\\\\\boxed{\sf g\big(f(x)\big)~=~6x~+~7}$

a)

Igualando a composta fog(x) a 0, temos:

$\sf 6x~-~4~=~0\\\\6x~=~0~+~4\\\\6x~=~4\\\\x~=~\dfrac{4}{6}\\\\Simplificando~a~fracao~por~2\\\\\boxed{\sf x~=~\dfrac{2}{3}}$

b)

Igualando a composta gof(x) a 1, temos:

$\sf 6x~+~7~=~1\\\\6x~=~1~-~7\\\\6x~=\,-6\\\\x~=~\dfrac{-6}{6}\\\\\boxed{\sf x~=\,-1}$

$\Huge{\begin{array}{c}\Delta \tt{\!\!\!\!\!\!\,\,o}\!\!\!\!\!\!\!\!\:\,\perp\end{array}}Qualquer~d\acute{u}vida,~deixe~ um~coment\acute{a}rio$