find the inverse of cube root of 2x-5

Answer 1

To find the inverse of the function \( y = \sqrt[3]{2x - 5} \), we follow these steps:

1. Replace \( y \) with \( f(x) \): \[ y = \sqrt[3]{2x - 5} \]

2. Switch \( x \) and \( y \): \[ x = \sqrt[3]{2y - 5} \]

3. Solve for \( y \): To eliminate the cube root, cube both sides: \[ x^3 = 2y - 5 \]

   Now, isolate \( y \): \[ 2y = x^3 + 5 \] \[ y = \frac{x^3 + 5}{2} \]

4. Final expression for the inverse: Thus, the inverse function is: \[ f^{-1}(x) = \frac{x^3 + 5}{2} \]

So, the inverse of \( \sqrt[3]{2x - 5} \) is \( \frac{x^3 + 5}{2} \).