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To find the inverse function \( f^{-1}(x) \) for the given function \( f(x) = 10 + \sqrt[3]{x} \), follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
Let:
\[
y = 10 + \sqrt[3]{x}
\]

### Step 2: Swap \( x \) and \( y \)
For the inverse function, switch the roles of \( x \) and \( y \):
\[
x = 10 + \sqrt[3]{y}
\]

### Step 3: Solve for \( y \)
Isolate \( y \):
\[
x - 10 = \sqrt[3]{y}
\]
Now, cube both sides to eliminate the cube root:
\[
(x - 10)^3 = y
\]

Thus, the inverse function is:
\[
f^{-1}(x) = (x - 10)^3
\]

Would you like further clarification or details?

Here are some related questions to practice:
1. What is the domain and range of the inverse function \( f^{-1}(x) \)?
2. How would you verify if the inverse function is correct?
3. What is the inverse of \( f(x) = 10 + \sqrt[4]{x} \)?
4. Can a function always have an inverse? Why or why not?
5. How would you graph the function \( f(x) \) and its inverse?

**Tip:** To find the inverse of a function, always swap \( x \) and \( y \), then solve for \( y \).

Find the inverse function of f(x) = 10 + ∛x.

Learn how to find the inverse function of f(x) = 10 + Cube Root of x. This problem explores key algebraic concepts related to inverse functions and cube roots, suitable for high school math students (Grades 9-12).

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