Question

find the inverse of the following function.
$f(x) = 2x^3 - 5$

• $f^{-1}(x) = sqrt3{\frac{x + 5}{2}}$
• $f^{-1}(x) = -2x^3 + 5$
• $f^{-1}(x) = sqrt3{\frac{x - 2}{5}}$
• $f^{-1}(x) = 5x^3 - 2$

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start by writing the function as \( y = 2x^3 - 5 \).

Step2: Swap \( x \) and \( y \)

To find the inverse, we interchange \( x \) and \( y \), so we get \( x = 2y^3 - 5 \).

Step3: Solve for \( y \)

First, add 5 to both sides of the equation: \( x + 5 = 2y^3 \). Then, divide both sides by 2: \( \frac{x + 5}{2} = y^3 \). Finally, take the cube root of both sides to solve for \( y \): \( y=\sqrt[3]{\frac{x + 5}{2}} \). Since \( y = f^{-1}(x) \), we have \( f^{-1}(x)=\sqrt[3]{\frac{x + 5}{2}} \).

Answer:

\( f^{-1}(x)=\sqrt[3]{\frac{x + 5}{2}} \) (the first option)