Find its inverse. State the domain and the range of f and f^-1. Graph f, f^-1, and y=x on the same coordinate axes.

f (x)= (x^3) + 1

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- Dec 16th 2008, 09:04 PMkukid123Functions and their graphs
Find its inverse. State the domain and the range of f and f^-1. Graph f, f^-1, and y=x on the same coordinate axes.

f (x)= (x^3) + 1 - Dec 16th 2008, 09:50 PMProve It
Unless otherwise stated, the domain of any polynomial is always $\displaystyle \mathbf{R}$.

It always helps to sketch the graph to start with.

So the domain of $\displaystyle f(x) = x^3 + 1$ is $\displaystyle \mathbf{R}$.

It's range is also $\displaystyle \mathbf{R}$.

To find the inverse, your x and y (or in this case, f) values swap, and so do the domain and range.

Since the domain and range of $\displaystyle f$ were both $\displaystyle \mathbf{R}$, so are the domain and range of $\displaystyle f^{-1}$.

Now we just find what the inverse function is.

$\displaystyle x = y^3 + 1$

$\displaystyle x - 1 = y^3$

$\displaystyle y = \sqrt[3]{x - 1}$

$\displaystyle f^{-1}(x) = \sqrt[3]{x - 1}$.

Now graph this function.