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If they've ever used a square root, they've used an inverse function!
Whenever they undo something that they or someone else did, they use an inverse function, whether it's untying a knot or solving a puzzle or decoding a secret message. When a computer reads a number you type in and converts it to binary for internal storage, then prints it out again on the screen for you to see, it's doing an inverse function.
If they know even a little trigonometry, they should know about the arctan function, the inverse of the tangent. And if they have used logarithms, they should know that a logarithm is the inverse of an exponential. These are used constantly in real life. (At least in the real life of people who use these sorts of math in their work, which is more than students realize.)
How do you define "real life"?
Does it have to be part of a kid's everyday experience outside of school, as opposed to something engineers do every day to design the devices kids use every day? I think "real life" is a lot bigger than most students realize! And most interesting applications of math occur in the context of other math, science, or engineering, rather than in the ordinary activities of life, so it's not realistic to pretend everyone will be using advanced math whenever they go shopping or something, or that anything outside their experience is unimportant. (This is what I wish I could say to all the students who write to us asking how topic X is used in real life, or to the teachers who apparently assign them to find out.)
I'll assume you are not trying to show the importance of inverse functions, but just looking for a way to relate the concept to something familiar.
How about this: When someone calls you on the phone, he or she looks up your number in a phone book (a function from names to phone numbers). When Caller ID shows who is calling, it has performed the inverse function, finding the name corresponding to the number.
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