If f : R → R be defined by f(x) = x^4, write f^ -1 (1). - Vidyadip

Question

If $f : R → $R be defined by $f(x) = x^4$, write $f^{-1}(1).$

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Answer

$ \text { Let } f^{-1}(1)=x \ldots . . .(1) $
$ \Rightarrow f(x)=1 $
$ \Rightarrow x^4=1 $
$ \Rightarrow x^4-1=0 $
$ \Rightarrow\left(x^2-1\right)\left(x^2+1\right)=0\left[\text { Using identity: } a^2-b^2=(a-b)(a+b)\right] $
$ \Rightarrow(x-1)(x+1)\left(x^2+1\right)=0\left[\text { Using identity: } a^2-b^2=(a-b)(a+b)\right] $
$ \Rightarrow x= \pm 1[\text { as } x \in R] $
$ \Rightarrow f^{-1}(1)=\{-1,1\}[\text { from (1)] }$

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