Question

# If $$f:R\rightarrow R$$ is given by $$f(x)={x}^{3}$$, write $${f}^{-1}(1)$$.

Solution

## $$f:R\rightarrow R$$ is given by $$f(x)=x^3$$.Let $$f^{-1}(1)=x$$                  ---- ( 1 )$$\Rightarrow$$  $$f(x)=1$$$$\Rightarrow$$  $$x^3=1$$  ...................[ Given ]$$\Rightarrow$$  $$x^3-1=0$$$$\Rightarrow$$  $$(x-1)(x^2+x+1)=0$$ ...................[ Since, $$a^3-b^3=(a+b)(a^2+ab+b^2)$$ ]Value of $$x$$ of $$x^2+x+1$$ will be complex number.As $$x\in R$$, then $$x^2+x+1=0$$ is not possible.$$\Rightarrow$$  $$x-1=0$$$$\Rightarrow$$  $$x=1$$From ( 1 ),$$\Rightarrow$$  $$f^{-1}(1)=\{1\}$$Mathematics

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