Question

Seja f uma função de z em z definida por (f(x)=4x-2)/3 (tudo sobre 3).
Em cada caso, determine, se existir, o número inteiro cuja imagem vale:
A)6
B)-10
C)0
D)1

Answer 1

$f(x)=\dfrac{4x-2}{3}~~~~\text{com }x\in\mathbb{Z}~\text{ e }~f(x)\in \mathbb{Z}$


Verificar se existe $x\in\mathbb{Z}\,,$ de forma que


A) $f(x)=6:$

$\dfrac{4x-2}{3}=6\\\\\\ 4x-2=6\cdot 3\\\\ 4x-2=18\\\\ 4x=18+2\\\\ 4x=20\\\\ x=\dfrac{20}{4}\\\\\\ \boxed{\begin{array}{c}x=5 \end{array}}\in\mathbb{Z}~~~~~~(\checkmark)$


B) $f(x)=-10:$

$\dfrac{4x-2}{3}=-10\\\\\\ 4x-2=-10\cdot 3\\\\ 4x-2=-30\\\\ 4x=-30+2\\\\ 4x=-28\\\\ x=\dfrac{-28}{4}\\\\\\ \boxed{\begin{array}{c}x=-7 \end{array}}\in\mathbb{Z}~~~~~~(\checkmark)$


C) $f(x)=0:$

$\dfrac{4x-2}{3}=0\\\\\\ 4x-2=0\\\\ 4x=2\\\\ x=\dfrac{2}{4}\not \in \mathbb{Z}~~~~~~(\diagup\!\!\!\!\!\diagdown)$


Logo, $0\not \in\mathrm{Im}(f)\,,$ pois não existe $x$ inteiro de forma que $f(x)=0.$


D) $f(x)=1:$

$\dfrac{4x-2}{3}=1\\\\\\ 4x-2=1\cdot 3\\\\\\ 4x-2=3\\\\ 4x=3+2\\\\ 4x=5\\\\ x=\dfrac{5}{4}\not\in\mathbb{Z}~~~~~~(\diagup\!\!\!\!\!\diagdown)$


Logo, $1\not\in\mathrm{Im}(f)\,,$ pois não existe $x$ inteiro de forma que $f(x)=1.$


Bons estudos! :-)