The function $$ f(x)=(x+1)^{3} $$ 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 $$ f\left(f^{-1}(x)\right)=x $$ and $$ f^{-1}(f(x))=x $$. A. $$ f^{-1}(x)=\square $$, for $$ x \neq \ldots $$ B. $$ f^{-1}(x)= $$, for $$ x 5^{\text {nan }} $$ C. $$ f^{-1}(x)=\sqrt[3]{x}-1 $$, for all $$ x $$ D. $$ f^{-1}(x)= $$ $$ \square $$ , $$ \operatorname{for} \mathrm{x} \geq $$ $$ \square $$ b. Verify that the equation is correct. $$ \begin{array}{rlrlrl} f\left(f^{-1}(x)\right) & =f(\sqrt[3]{x}-1) & \text { and } & f^{-1}(f(x)) & =f^{-1}(\square) & \ & =x & & \text { Substitute. } \ & =x & & \text { Simplify. } \end{array} $$

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Let $$ f(x) = (x+1)^3 $$. We want to find $$ f^{-1}(x) $$ by following these steps:

1. Let $$ y = f(x) $$. Then,  
   $$
   y = (x+1)^3.
   $$

2. Solve for $$ x $$ in terms of $$ y $$:  
   Take the cube root of both sides:
   $$
   \sqrt[3]{y} = x+1.
   $$
   Then,
   $$
   x = \sqrt[3]{y} - 1.
   $$

3. Replace $$ y $$ with $$ x $$ to write the inverse function:
   $$
   f^{-1}(x) = \sqrt[3]{x} - 1,
   $$
   which is valid for all $$ x $$.

4. Verification:

- Compute $$ f(f^{-1}(x)) $$:  
     $$
     f\bigl(f^{-1}(x)\bigr) = f\left(\sqrt[3]{x} - 1\right) = \left[(\sqrt[3]{x} - 1) + 1\right]^3 = (\sqrt[3]{x})^3 = x.
     $$

- Compute $$ f^{-1}(f(x)) $$:  
     $$
     f^{-1}(f(x)) = f^{-1}\bigl((x+1)^3\bigr) = \sqrt[3]{(x+1)^3} - 1 = x+1 - 1 = x.
     $$

Thus, the inverse function is correctly given by  
$$
f^{-1}(x) = \sqrt[3]{x} - 1, \quad \text{for all } x.
$$
$$ f^{-1}(x) = \sqrt[3]{x} - 1 $$, for all $$ x $$. Verification: $$ f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x $$

The function is one-to-one.
a. Find an equation for , the inverse function.
b. Verify that your equation is correct by showing that and .
A. , for
B. , for
C. , for all
D. ,
b. Verify that the equation is correct.

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The function is one-to-one. a. Find an equation for , the inverse function. b. Verify that your equation is correct by showing that and . A. , for B. , for C. , for all D. , b. Verify that the equation is correct.