Question

Sejam feg funções de R em R dadas por g(x) = 3x-2
e (fºg)(x) = 9x^2 – 3x + 1. Qual é a lei da função f?

Answer 1

$\Large\boxed{\begin{array}{l}\sf g(x)=3x-2\longrightarrow x=\dfrac{g(x)+2}{3}\\\sf f\circ g(x)=9x^2-3x+1\\\sf f[g(x)]=9x^2-3x+1\\\sf f[g(x)]=9\cdot\bigg(\dfrac{g(x)+2}{3}\bigg)^2-3\cdot\dfrac{g(x)+2}{3}+1\\\sf f[g(x)]=\backslash\!\!\!9\cdot\dfrac{g(x)^2+4g(x)+4}{\backslash\!\!\!9}-\backslash\!\!\!3\cdot\dfrac{g(x)+2}{\backslash\!\!\!3}+1\\\sf f[g(x)]=g(x)^2+4g(x)+4-g(x)-2+1\\\sf f[g(x)]= g(x)^2+3g(x)+3\\\huge\boxed{\boxed{\boxed{\boxed{\sf f(x)= x^2+3x+3}}}}\end{array}}$