What procedure is available to determine if two functions are inverses of each other?

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

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Graphically, two functions are inverses of each other if they're mirror images across the line `y=x.` For example, if `f(x)=x^3` and `g(x)=root(3)(x),` then `f` and `g` are inverses of each other. Here are their graphs, and the graph of `y=x` in red:

As you can see, they're mirror images with respect to the red line. This is a helpful way to view inverse functions, but sometimes it can be hard or impossible to tell if the graphs are mirror images by eye.

The algebraic way to check whether two functions are inverses is to compose the two with each other and see if the resulting function is the identity function. In other words, if

`f(g(x))=x` and `g(f(x))=x,`

then `f` and `g` are inverses. If we use the same example as above, with `f(x)=x^3` and `g(x)=root(3)(x),` we get

`f(g(x))=f(root(3)(x))=(root(3)(x))^3=x,` and

`g(f(x))=g(x^3)=root(3)(x^3)=x,` so now we've shown algebraically that the two functions are inverses.

I'd recommend solving problems both ways. Either graph the two functions first to verify that they're mirror images across `y=x` and then compute `f(g(x))` and `g(f(x)),` or the other way around. This gives you practice both ways and lets you check your work.


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