Everyday usage and use in mathematics
In everyday English, The inverse of something is an opposite. The Oxford English Dictionary[1] defines inverse as "The opposite or contrary of something; esp. a situation, process, state of affairs, etc., which is the reverse of that which has been previously stated or described." For example, you might say "At age 5, Sarah was taller than Jane; at age 8 the inverse was true." This means that age 8, Jane was taller than Sarah.
In mathematics, an inverse function is a function that "undoes" another function. For example, the inverse of the function $f(x) = 3x + 9$ is $f^{-1}(y) = \frac{y - 9}{3}$. squaring a number is taking the square root. A function that has an inverse is called an invertible function.
The mathematical term inverse function is related to the everyday definition of the word inverse in the case of a process or a procedure. For example, the inverse of tying your shoes is untying your shoes. If you untie your shoes, this "undoes" the process of tying your shoes. If you square a positive number and then take the square root of the result, you get the original number back.
Summary
In everyday english, an inverse is an opposite situation or process. In mathematics, an inverse function "undoes" a function.
Importance in mathematics
Inverse functions are important for solving problems where you're presented the "opposite" of the usual type of information you're given in a problem. For example, let's say you know that a dropped ball is at height $h = f(t)$ at time $t$. If you need to figure out time $t$ when the ball reaches a certain height $h$, you need to find the inverse of a function $f$. The inverse of the function $f$ is usually written as $f^{-1}$.
Certain inverse functions are so important that they are given special names. The inverse of an exponential function is a logarithmic function. The inverse of the sine function is the arcsine function. In order for a function to be invertible, it must be a one-to-one function. For example, the function $g(x) = x^2$ is not one-to-one on the domain $(-\infty, \infty)$ because $g(-3) = g(3) = 9$. Hence, there can't be a well-defined value for $g^{-1}(9)$: it's not clear whether it should be $3$ or $-3$.
Exercises
- In everyday english, what is the inverse of driving to work?
- Suppose $f$ is an invertible function and that $f(3) = 5$. What is $f^{-1}(5)$?
References
[1] Oxford English Dictionary