The function $$ f(x)=x^{3}+10 $$ is one-to-one. a. Find an equation for $$ f^{-1} $$, 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 $$ x \geq \square $$ B. $$ f^{-1}(x)=\square $$, for $$ x
eq \square $$ C. $$ f^{-1}(x)=\square $$, for $$ x \leq \square $$ b. Verify that the equation is correct. $$ f^{-1}(x)=\square $$, for all $$ x $$ $$ f\left(f^{-1}(x) ight)=f(\square) $$

Question

The function $$ f(x)=x^{3}+10 $$ is one-to-one. a. Find an equation for $$ f^{-1} $$, 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 $$ x \geq \square $$ B. $$ f^{-1}(x)=\square $$, for $$ x 
eq \square $$ C. $$ f^{-1}(x)=\square $$, for $$ x \leq \square $$ b. Verify that the equation is correct. $$ f^{-1}(x)=\square $$, for all $$ x $$ $$ f\left(f^{-1}(x)ight)=f(\square) $$

UpStudy AI · Accepted Answer

To find the inverse function $$ f^{-1}(x) $$ for the function $$ f(x) = x^3 + 10 $$, we will follow these steps:

### Part a: Finding the Inverse Function

1. **Set the function equal to $$ y $$**:
   $$
   y = x^3 + 10
   $$

2. **Solve for $$ x $$** in terms of $$ y $$:
   $$
   y - 10 = x^3
   $$
   $$
   x = \sqrt[3]{y - 10}
   $$

3. **Express the inverse function**:
   $$
   f^{-1}(x) = \sqrt[3]{x - 10}
   $$

4. **Determine the domain of the inverse function**:
   Since $$ f(x) $$ is a cubic function, it is defined for all real numbers. Therefore, the inverse function $$ f^{-1}(x) $$ is also defined for all real numbers.

Thus, we can fill in the answer box for part a:
- **Choice A**: $$ f^{-1}(x) = \sqrt[3]{x - 10} $$, for $$ x \in \mathbb{R} $$

### Part b: Verification of 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 $$ f(x) $$:
$$
f(f^{-1}(x)) = f\left(\sqrt[3]{x - 10}\right)
$$
Calculating $$ f\left(\sqrt[3]{x - 10}\right) $$:
$$
= \left(\sqrt[3]{x - 10}\right)^3 + 10
$$
$$
= x - 10 + 10
$$
$$
= x
$$

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

Substituting $$ f(x) $$ into $$ f^{-1}(x) $$:
$$
f^{-1}(f(x)) = f^{-1}(x^3 + 10)
$$
Calculating $$ f^{-1}(x^3 + 10) $$:
$$
= \sqrt[3]{(x^3 + 10) - 10}
$$
$$
= \sqrt[3]{x^3}
$$
$$
= x
$$

Both verifications confirm that the inverse function is correct.

### Final Answers

- **Part a**: $$ f^{-1}(x) = \sqrt[3]{x - 10} $$, for $$ x \in \mathbb{R} $$
- **Part b**: $$ f^{-1}(x) = \sqrt[3]{x - 10} $$, for all $$ x $$ and $$ f(f^{-1}(x)) = x $$ and $$ f^{-1}(f(x)) = x $$ are verified.
a. The inverse function is $$ f^{-1}(x) = \sqrt[3]{x - 10} $$, for all real numbers $$ x $$. b. Verification: - $$ f(f^{-1}(x)) = x $$ - $$ f^{-1}(f(x)) = x $$

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