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For the function f(x)=root(3)((4x)), find f^(-1)(x). f^(-1)(x)=((x)/(4))^(3) f^(-1)(x)=(4x)^(3) f^(-1)(x)=(x^(3))/(4) f^(-1)(x)=4x^(3)

For the function f(x)=(4x)3 f(x)=\sqrt[3]{(4 x)} , find f−1(x) f^{-1}(x) .\newlinef−1(x)=(x4)3 f^{-1}(x)=\left(\frac{x}{4}\right)^{3} \newlinef−1(x)=(4x)3 f^{-1}(x)=(4 x)^{3} \newlinef−1(x)=x34 f^{-1}(x)=\frac{x^{3}}{4} \newlinef−1(x)=4x3 f^{-1}(x)=4 x^{3}

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Simplify the product of two radical expressions having same variableright chevron icon

Full solution

Q. For the function f(x)=(4x)3 f(x)=\sqrt[3]{(4 x)} , find f−1(x) f^{-1}(x) .\newlinef−1(x)=(x4)3 f^{-1}(x)=\left(\frac{x}{4}\right)^{3} \newlinef−1(x)=(4x)3 f^{-1}(x)=(4 x)^{3} \newlinef−1(x)=x34 f^{-1}(x)=\frac{x^{3}}{4} \newlinef−1(x)=4x3 f^{-1}(x)=4 x^{3}
  1. Write function as yy: To find the inverse function, we first write the function as y=4x3y = \sqrt[3]{4x}.
  2. Express cube root as exponent: Next, we express the cube root as an exponent: y=(4x)13y = (4x)^{\frac{1}{3}}.
  3. Swap xx and yy: To find the inverse, we swap xx and yy, so we get x=(4y)13x = (4y)^{\frac{1}{3}}.
  4. Cube both sides: Now we need to solve for yy. To do this, we cube both sides of the equation to get rid of the cube root: x3=4yx^3 = 4y.
  5. Isolate yy: Divide both sides by 44 to isolate yy: y=x34y = \frac{x^3}{4}.
  6. Find inverse function: Now we have the inverse function, which we denote as f−1(x)f^{-1}(x): f−1(x)=x34f^{-1}(x) = \frac{x^3}{4}.

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