How do you find the inverse function of #f(x)=x^3+5#?

1 Answer
Bill K.
Sep 24, 2015

#f^{-1}(x)=root(3)(x-5)#

Explanation:

Set #y=f(x)=x^3+5# and solve for #x#: #x^3=y-5\rightarrow x=root(3)(y-5)#.

It's okay to write the answer as #x=f^{-1}(y)=root(3)(y-5)#. It's also okay to swap variables and write the answer as #y=f^{-1}(x)=root(3)(x-5)#.

Swapping the variables is a good idea when you want to graph both functions in the same picture while seeing the "reflection property" of a function and its inverse across the line #y=x#. Not swapping the variables is a good idea if they have a real life meaning (which you would "lose" by swapping them).

Do note that the original function is indeed a one-to-one function since its graph passes the horizontal line test (see the graph below). There's no need to restrict its domain in any way.

graph{x^3+5 [-40, 40, -20, 20]}

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