Question

If f : R  R defined by f (x) = x3

– 1, then f–1 {– 2, 0, 7| = ___​

Answer 1

$\textbf{Given:}$

$f:\,R\implies\,R\;\text{by}\;f(x)=x^3-1$

$\textbf{To find:}\;f^{-1}\{-2,0,7\}$

$\textbf{Solution:}$

$\bf\,f^{-1}\{-2,0,7\}$

$=\text{Set of all pre images of -2, 0 and 7}$

$\text{Now,}$

$f(x)=-2$

$\implies\,x^3-1=-2$

$\implies\,x^3=-2+1$

$\implies\,x^3=-1$

$\implies\bf\,x=-1$

$\therefore\textbf{Pre image of -2 is -1}$

$f(x)=0$

$\implies\,x^3-1=0$

$\implies\,x^3=1$

$\implies\bf\,x=1$

$\therefore\textbf{pre image of 0 is 1}$

$f(x)=7$

$\implies\,x^3-1=7$

$\implies\,x^3=1+7$

$\implies\,x^3=8$

$\implies\bf\,x=2$

$\therefore\textbf{pre image of 7 is 2}$

$\text{Hence,}\;\bf\,f^{-1}\{-2,0,7\}=\{-1,1,2\}$

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