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Question Number 12855 by kunalshukla95040 last updated on 04/May/17

$$f\left(x\right)={[1-{(x-3)}^4]}^{1/7}$$$$find{f}^{-1}\left(x\right).$$

Answered by linkelly0615 last updated on 04/May/17

$$f\left(x\right)={[1-{(x-3)}^4]}^{1/7}$$$$\Rightarrow {\left(f\left(x\right)\right)}^7=1-{(x-3)}^4$$$$\Rightarrow 1-{\left(f\left(x\right)\right)}^7={(x-3)}^4$$$$\Rightarrow {[1-{\left(f\left(x\right)\right)}^7]}^{1/4}=x-3$$$$\Rightarrow {[1-{\left(f\left(x\right)\right)}^7]}^{1/4}+3=x$$$$\because f^{-1}\left(f\left(x\right)\right)=x$$$$\therefore f^{-1}\left(x\right)={[1-x^7]}^{1/4}+3$$

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