Question

find the inverse of cube of 2x-5

Answer 1

To find the inverse of the function \( f(x) = (2x - 5)^3 \), we follow these steps:

1. **Replace \( f(x) \) with \( y \)**:
\[
y = (2x - 5)^3
\]

2. **Switch \( x \) and \( y \)** to find the inverse:
\[
x = (2y - 5)^3
\]

3. **Solve for \( y \)**:
- Take the cube root of both sides:
\[
\sqrt[3]{x} = 2y - 5
\]

- Add 5 to both sides:
\[
\sqrt[3]{x} + 5 = 2y
\]

- Divide by 2:
\[
y = \frac{\sqrt[3]{x} + 5}{2}
\]

4. **Write the inverse function**:
\[
f^{-1}(x) = \frac{\sqrt[3]{x} + 5}{2}
\]

So, the inverse of the cube of \( 2x - 5 \) is:
\[
f^{-1}(x) = \frac{\sqrt[3]{x} + 5}{2}
\]