Find the Inverse of a Cubic Function

A cubic function of the form \( f(x) = a(px + q)^3 + r \) has an inverse that can be found by solving for \( x \) in terms of \( y \).

The inverse function is given by:

\( f^{-1}(x) = \frac{1}{p} \left( \sqrt[3]{\frac{x - r}{a}} - q \right) \)

All coefficients are kept as exact fractions and cube roots are simplified when possible.

✧ Inverse of a Cubic Function ✧

Enter coefficients for f(x) = a(px + q)³ + r

Function: \( f(x) = a(px + q)^3 + r \)

\( f(x) = \) \( 2(x - 2)^3 + 3 \)

⚠️ If p = 0, it will be set to 1. Exact fractions and simplified cube roots.
📐 Inverse Function \( f^{-1}(x) \)
\( f^{-1}(x) = \sqrt[3]{\frac{x - 3}{2}} + 2 \)

📖 Step-by-Step Solution

STEP 1: Write the function as an equation replacing \( f(x) \) by \( y \)
\( y = 2(x - 2)^3 + 3 \)
STEP 2: Solve for \( x \) in terms of \( y \)

\( y = 2(x - 2)^3 + 3 \)

\( y - 3 = 2(x - 2)^3 \)

\( \frac{y - 3}{2} = (x - 2)^3 \)

\( \sqrt[3]{\frac{y - 3}{2}} = x - 2 \)

\( x = \sqrt[3]{\frac{y - 3}{2}} + 2 \)

STEP 3: Interchange \( x \) and \( y \) to get the inverse
\( y = \sqrt[3]{\frac{x - 3}{2}} + 2 \)
STEP 4: Write the inverse function notation
\( f^{-1}(x) = \sqrt[3]{\frac{x - 3}{2}} + 2 \)
💡 Interpretation
The inverse function reverses the effect of the original function. The domain of \( f \) becomes the range of \( f^{-1} \), and vice versa. Cubic functions are one-to-one, so their inverses are also functions.
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