Question

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Part 1
The function f left parenthesis x right parenthesis equals x cubed minus 6 is​ one-to-one.
a. Find an equation for f Superscript negative 1​, the inverse function.
b. Verify that your equation is correct by showing that f left parenthesis f Superscript negative 1 Baseline left parenthesis x right parenthesis right parenthesisequalsx and f Superscript negative 1 Baseline left parenthesis f left parenthesis x right parenthesis right parenthesisequalsx.
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Part 1
a. Select the correct choice below and fill in the answer​ box(es) to complete your choice.
​(Simplify your answer. Use integers or fractions for any numbers in the​ expression.)
A.
f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here​, for xless than or equals
  
enter your response here
B.
f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
RootIndex 3 StartRoot x plus 6 EndRoot​, for all x
Your answer is correct.C.
f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here​, for xgreater than or equals
  
enter your response here
D.
f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here​, for xnot equals
  
enter your response here
Part 2
b. Verify that the equation is correct.
f left parenthesis f Superscript negative 1 Baseline left parenthesis x right parenthesis right parenthesis
equals
fleft parenthesis nothing right parenthesis
  
and
f Superscript negative 1 Baseline left parenthesis f left parenthesis x right parenthesis right parenthesis
equals
f Superscript negative 1 Baseline left parenthesis nothing right parenthesis
  
Substitute.
equals
  
enter your response here
equals
  
enter your response here
Simplify.

b. Verify that the equation is correct.
f left parenthesis f Superscript negative 1 Baseline left parenthesis x right parenthesis right parenthesis
equals
fleft parenthesis nothing right parenthesis
  
and
f Superscript negative 1 Baseline left parenthesis f left parenthesis x right parenthesis right parenthesis
equals
f Superscript negative 1 Baseline left parenthesis nothing right parenthesis
  
Substitute.
equals
  
enter your response here
equals
  
enter your response here
Simplify.

Answer 1

To find the inverse function of \( f(x) = x^3 - 6 \), we will follow the steps to isolate \( x \) and express the function in terms of \( y \).

### Part 1: Finding the Inverse Function

1. **Rewrite the function**:
Let \( y = f(x) = x^3 - 6 \).

2. **Solve for \( x \)**:
\[
y + 6 = x^3
\]
\[
x = \sqrt[3]{y + 6}
\]

3. **Express in terms of \( x \)**:
The inverse function is:
\[
f^{-1}(x) = \sqrt[3]{x + 6}
\]

Thus, the answer for Part 1, option B is:
\[
f^{-1}(x) = \sqrt[3]{x + 6}, \text{ for all } x
\]

### Part 2: Verification

We need to verify that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

1. **Verify \( f(f^{-1}(x)) = x \)**:
\[
f(f^{-1}(x)) = f(\sqrt[3]{x + 6})
\]
Substitute \( \sqrt[3]{x + 6} \) into the original function:
\[
= \left(\sqrt[3]{x + 6}\right)^3 - 6
\]
Simplifying gives:
\[
= (x + 6) - 6 = x
\]

2. **Verify \( f^{-1}(f(x)) = x \)**:
\[
f^{-1}(f(x)) = f^{-1}(x^3 - 6)
\]
Substitute \( x^3 - 6 \) into the inverse function:
\[
= \sqrt[3]{(x^3 - 6) + 6}
\]
Simplifying gives:
\[
= \sqrt[3]{x^3} = x
\]

Both verifications lead to \( x \), confirming that the inverses work as expected.

### Final Answers
A:
\[
f^{-1}(x) = \sqrt[3]{x + 6}, \text{ for all } x
\]
B:
The verification shows:
\[
f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x
\]