Finding all inverses Find all the inverses associated with the following functions, and state their domains.


ƒ(x) = (x + 1)³

Verified step by step guidance

1. Step 1: To find the inverse of the function $f(x)=(x+1{)}^3$, start by replacing $f(x)$ with $y$. This gives us the equation $y=(x+1{)}^3$.
2. Step 2: Swap the variables $x$ and $y$ to begin solving for the inverse. This results in the equation $x=(y+1{)}^3$.
3. Step 3: Solve for $y$ by taking the cube root of both sides to isolate $y+1$. This gives $\sqrt[3]{x}=y+1$.
4. Step 4: Isolate $y$ by subtracting 1 from both sides of the equation. This results in $y=\sqrt[3]{x}-1$.
5. Step 5: The inverse function is $f^{-1}(x)=\sqrt[3]{x}-1$. The domain of the inverse function is all real numbers, $\mathbb{R}$, since the cube root function is defined for all real numbers.