Question

Seja f a função de ℝ em ℝ dada por f(x)=-2x. Um esboço gráfico da função inversa de f é: *

Answer 1

$\large\boxed{\begin{array}{l}\sf f(x)=-2x\\\sf -2x=f(x)\\\sf x=-\dfrac{f(x)}{2}\\\\\sf f^{-1}(x)=-\dfrac{x}{2}\\\underline{\rm Demonstrac_{\!\!,}\tilde ao\!:}\\\sf f\circ f^{-1}(x)=-2\cdot f^{-1}(x)_\\\sf f\circ f^{-1}(x)=-\diagup\!\!\!2\cdot\bigg(-\dfrac{x}{\diagup\!\!\!2}\bigg)=-(-x)=x(func_{\!\!,}\tilde ao\,identidade.)\\\sf Analogamente,\\\sf f^{-1}\circ f(x)=-\dfrac{f(x)}{2}\\\\\sf f\circ f^{-1}(x)=-\dfrac{-\backslash\!\!\!2x}{\backslash\!\!\!2}=-(-x)=x.\end{array}}$

$\large\boxed{\begin{array}{l}\sf Se\,a\,composta\,de\,duas\,func_{\!\!,}\tilde oes\,resultar\\\sf na\,func_{\!\!,}\tilde ao\,identidade\\\sf ent\tilde ao\,uma\,\acute e\,inversa\,da\,outra.\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\tt O\,gr\acute afico\,est\acute a\,anexo}\\\boxed{\begin{array}{c|c}\sf x&\sf f^{-1}(x)\\\sf -4&\sf2\\\sf-3&\sf\dfrac{3}{2}\\\\\sf -1&\sf\dfrac{1}{2}\\\\\sf 0&\sf0\\\sf 1&\sf-\dfrac{1}{2}\\\\\sf 2&-1\end{array}}\end{array}}$