The function $$ f(x)=(x+9)^{3} $$ is one-to-one. a. Find an equation for $$ f^{-1}(x) $$, the inverse function. b. Verify that your equation is correct by showing that $$ f\left(f^{-1}(x) ight)=x $$ and $$ f^{-1}(f(x))=x $$. 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. $$ f^{-1}(x)=\square $$, for $$ x
eq \square $$ B. $$ f^{-1}(x)=\square $$, for $$ x \leq \square $$ D. $$ f^{-1}(x)=\square $$, for all $$ x $$ b. Verify that the equation is correct. $$ f\left(f^{-1}(x) ight)=f(\square) $$

Question

The function $$ f(x)=(x+9)^{3} $$ is one-to-one. a. Find an equation for $$ f^{-1}(x) $$, the inverse function. b. Verify that your equation is correct by showing that $$ f\left(f^{-1}(x)ight)=x $$ and $$ f^{-1}(f(x))=x $$. 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. $$ f^{-1}(x)=\square $$, for $$ x 
eq \square $$ B. $$ f^{-1}(x)=\square $$, for $$ x \leq \square $$ D. $$ f^{-1}(x)=\square $$, for all $$ x $$ b. Verify that the equation is correct. $$ f\left(f^{-1}(x)ight)=f(\square) $$

UpStudy AI · Accepted Answer

To find the inverse function $$ f^{-1}(x) $$ for the function $$ f(x) = (x + 9)^3 $$, we will follow these steps:

### Part a: Finding the Inverse Function

1. **Start with the function**:
   $$
   y = (x + 9)^3
   $$

2. **Swap $$ x $$ and $$ y $$** to find the inverse:
   $$
   x = (y + 9)^3
   $$

3. **Solve for $$ y $$**:
   - Take the cube root of both sides:
   $$
   y + 9 = \sqrt[3]{x}
   $$
   - Subtract 9 from both sides:
   $$
   y = \sqrt[3]{x} - 9
   $$

4. **Write the inverse function**:
   $$
   f^{-1}(x) = \sqrt[3]{x} - 9
   $$

Now, we can fill in the answer box for part a. The correct choice is:
- **D.** $$ f^{-1}(x) = \sqrt[3]{x} - 9 $$, for all $$ x $$

### Part b: Verifying the Inverse Function

To verify that our inverse function is correct, we need to show that:
1. $$ f(f^{-1}(x)) = x $$
2. $$ f^{-1}(f(x)) = x $$

#### Step 1: Verify $$ f(f^{-1}(x)) = x $$

Substituting $$ f^{-1}(x) $$ into $$ f(x) $$:
$$
f(f^{-1}(x)) = f(\sqrt[3]{x} - 9)
$$
Calculating $$ f(\sqrt[3]{x} - 9) $$:
$$
= ((\sqrt[3]{x} - 9) + 9)^3
$$
$$
= (\sqrt[3]{x})^3
$$
$$
= x
$$

#### Step 2: Verify $$ f^{-1}(f(x)) = x $$

Substituting $$ f(x) $$ into $$ f^{-1}(x) $$:
$$
f^{-1}(f(x)) = f^{-1}((x + 9)^3)
$$
Calculating $$ f^{-1}((x + 9)^3) $$:
$$
= \sqrt[3]{(x + 9)^3} - 9
$$
$$
= (x + 9) - 9
$$
$$
= x
$$

Both verifications confirm that the inverse function is correct.

### Final Answers
- a. **D.** $$ f^{-1}(x) = \sqrt[3]{x} - 9 $$, for all $$ x $$
- b. $$ f(f^{-1}(x)) = f(\sqrt[3]{x} - 9) = x $$ and $$ f^{-1}(f(x)) = f^{-1}((x + 9)^3) = x $$
a. The inverse function is $$ f^{-1}(x) = \sqrt[3]{x} - 9 $$ for all $$ x $$. b. - $$ f(f^{-1}(x)) = x $$ - $$ f^{-1}(f(x)) = x $$

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