Question

Find all the inverses associated with the following functions and state their domains. $$f(x)=(x+1)^{3}$$

Short answer

Answer: The inverse of the function $$f(x) = (x+1)^3$$ is $$f^{-1}(x) = \sqrt[3]{x} - 1$$, and its domain is $$x \in \mathbb{R}$$.

Step-by-step solution

Swap input and output of the function

To find the inverse of the function, replace f(x) with y and swapx and y. So, we will have: $$x = (y+1)^{3}$$

Step 2:mSolve for y

Now, solve for y by taking the cube root of both sides: $$\sqrt[3]{x} = y + 1$$ Subtract 1 from both sides to isolate y: $$y = \sqrt[3]{x} - 1$$

Write the inverse function

We found that the inverse function is given by: $$f^{-1}(x) = \sqrt[3]{x} - 1$$

Determine the domain of the inverse function

Since the cube root function can have any real number as an input, the domain of the inverse function is: $$x \in \mathbb{R}$$ So, the inverse of the function $$f(x) = (x+1)^{3}$$ is $$f^{-1}(x) = \sqrt[3]{x} - 1$$ with the domain $$x \in \mathbb{R}$$