Find the inverse function. $$ f(x)=\sqrt[3]{x}+1 $$ $$ f^{-1}(x)=(x+[?]) $$

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To find the inverse function $$ f^{-1}(x) $$ of the function $$ f(x) = \sqrt[3]{x} + 1 $$, follow these steps:

1. **Start with the original function:**
   $$
   y = \sqrt[3]{x} + 1
   $$
   
2. **Solve for $$ x $$ in terms of $$ y $$:**
   $$
   y - 1 = \sqrt[3]{x}
   $$
   $$
   (y - 1)^3 = x
   $$
   
3. **Express the inverse function:**
   $$
   f^{-1}(x) = (x - 1)^3
   $$

So, the inverse function is:
$$
f^{-1}(x) = (x - 1)^3
$$

**Verification:**

- **Compose $$ f $$ and $$ f^{-1} $$ to ensure they cancel out:**
  $$
  f(f^{-1}(x)) = \sqrt[3]{(x - 1)^3} + 1 = x - 1 + 1 = x
  $$
  $$
  f^{-1}(f(x)) = ( \sqrt[3]{x} + 1 - 1 )^3 = x
  $$

Both compositions return $$ x $$, confirming that $$ f^{-1}(x) $$ is indeed the inverse of $$ f(x) $$.

**Final Answer:**
$$
f^{-1}(x) = (x - 1)^3
$$
The inverse function is $$ f^{-1}(x) = (x - 1)^3 $$.

Find the inverse function. \( f(x)=\sqrt[3]{x}+1 \) \( f^{-1}(x)=(x+[?]) \)

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