For the function f(x)=root(5)(x^(7)-8), find f^(-1)(x). f^(-1)(x)=(root(7)(x))^(5)+8 f^(-1)(x)=root(7)(x^(5)+8) f^(-1)(x)=root(7)((x+8)^(5)) f^(-1)(x)=root(7)(x^(5))+8

Write function as y: To find the inverse function, we first write the function as y = root(5)(x^7 - 8).Express function in terms: Next, we express the function in terms of x: x = root(5)(y^7 - 8).Eliminate fifth root: Now we raise both sides of the equation to the power of 5 to eliminate the fifth root: x^5 = y^7 - 8.Isolate term with y: We then add 8 to both sides to isolate the term with y: x^5 + 8 = y^7.Solve for y: Next, we take the seventh root of both sides to solve for y: root(7)(x^5 + 8) = y.Express inverse function: Finally, we express the inverse function in terms of x: f^(-1)(x) = root(7)(x^5 + 8).

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Q. For the function

f(x)=\sqrt[5]{x^{7}-8}

, find

f^{-1}(x)

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f^{-1}(x)=(\sqrt[7]{x})^{5}+8

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f^{-1}(x)=\sqrt[7]{x^{5}+8}

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f^{-1}(x)=\sqrt[7]{(x+8)^{5}}

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f^{-1}(x)=\sqrt[7]{x^{5}}+8

Write function as $y$ : To find the inverse function, we first write the function as $y = \sqrt[5]{x^7 - 8}$ .
Express function in terms: Next, we express the function in terms of $x$ : $x = \sqrt[5]{y^7 - 8}$ .
Eliminate fifth root: Now we raise both sides of the equation to the power of $5$ to eliminate the fifth root: $x^5 = y^7 - 8$ .
Isolate term with y: We then add $8$ to both sides to isolate the term with $y$ : $x^5 + 8 = y^7$ .
Solve for $y$ : Next, we take the seventh root of both sides to solve for $y$ : $\sqrt[7]{x^5 + 8} = y$ .
Express inverse function: Finally, we express the inverse function in terms of $x$ : $f^{-1}(x) = \sqrt[7]{x^5 + 8}$ .