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Part 1

The function 

f left parenthesis x right parenthesis equals x cubed minus 6f(x)=x3−6 

is​ one-to-one.

Find an equation for 

f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x)​, 

the inverse function.

To find the inverse of the function \( f(x) = x^3 - 6 \), we need to follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
Let \( y = f(x) \), so:
\[
y = x^3 - 6
\]

### Step 2: Swap \( x \) and \( y \)
To find the inverse, swap \( x \) and \( y \):
\[
x = y^3 - 6
\]

### Step 3: Solve for \( y \)
Now solve for \( y \) in terms of \( x \):

\[
x + 6 = y^3
\]
\[
y = \sqrt[3]{x + 6}
\]

### Step 4: Write the inverse function
So the inverse function is:
\[
f^{-1}(x) = \sqrt[3]{x + 6}
\]

### Final Answer:
\[
f^{-1}(x) = \sqrt[3]{x + 6}
\]

Let me know if you would like further clarification on any steps, or details on how the function and its inverse relate!

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Here are five related questions you might find interesting:
1. How can you verify that \( f(x) = x^3 - 6 \) is one-to-one?
2. What is the domain and range of \( f(x) \) and \( f^{-1}(x) \)?
3. How do you check if two functions are inverses of each other?
4. Can the inverse function \( f^{-1}(x) \) have any restrictions in its domain?
5. How would this process change for more complicated functions like \( f(x) = x^3 - 6x + 2 \)?

**Tip**: When finding the inverse, remember that the graph of a function and its inverse are reflections of each other across the line \( y = x \).

The function f(x) = x^3 - 6 is one-to-one. Find an equation for f^(-1)(x), the inverse function.

Learn how to find the inverse of the cubic function f(x) = x^3 - 6 with step-by-step instructions. This guide covers key concepts such as inverse functions, one-to-one functions, and solving cubic equations. Suitable for students in Grades 10-12.

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