What is the inverse of the function f(x) = x^9 + 2?

To find the inverse of the function f(x) = x⁹ + 2, we first need to replace f(x) with y:

y = x⁹ + 2

Next, we switch x and y, since the inverse function essentially swaps the roles of x and y:

x = y⁹ + 2

Now, we solve for y. Start by isolating the term with y:

x – 2 = y⁹

Next, take the ninth root of both sides to solve for y:

y = (x – 2)^1/9

Thus, we can write the inverse function as:

f^-1(x) = (x – 2)^1/9

This means that for every output value from the original function, you can find the corresponding input value using the inverse function. The inverse function effectively undoes the operation performed by f(x).