For the function f(x)=frac{x^{frac{1}{3}}}{5}, find f^{-1}(x).

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Accepted Answer

Step:1 ：(f^{-1}(x)) is defined as the number (y) such that (f(y)=x). Thus, we solve the equation (frac{y^{frac{1}{3}}}{5} = x).Step:2 ：Multiplying both sides by 5, we have (y^{frac{1}{3}} = 5x).Step:3 ：Now, we raise both sides to the power of 3 to eliminate the cube root: ((y^{frac{1}{3}})^3 = (5x)^3).Step:4 ：This simplifies to (y = 125x^3).Step:5 ：So, the inverse function is (f^{-1}(x) = boxed{125x^3}).

Answer

Step: 1 ：(f^{-1}(x)) is defined as the number (y) such that (f(y)=x). Thus, we solve the equation (frac{y^{frac{1}{3}}}{5} = x).Step: 2 ：Multiplying both sides by 5, we have (y^{frac{1}{3}} = 5x).Step: 3 ：Now, we raise both sides to the power of 3 to eliminate the cube root: ((y^{frac{1}{3}})^3 = (5x)^3).Step: 4 ：This simplifies to (y = 125x^3).Step: 5 ：So, the inverse function is (f^{-1}(x) = boxed{125x^3}).

For the function $f(x)=\frac{x^{\frac{1}{3}}}{5}$, find $f^{-1}(x)$.

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