Here we “undo” functions.

**inverses**of each other if for all in , and also for all in . Sometimes we write in this case.

On the other hand,

So far, we’ve only dealt with abstract examples. Let’s see if we can ground this in a real-life context.

To find the inverse function, first note that Now write out the left-hand side of the equation and solve for .

So is the inverse function of , which converts a Fahrenheit measurement back into a Celsius measurement. The domain of this inverse function is .

Finally, we could check our work again using the definition of inverse functions. We have already guaranteed that since we solved for in our calculation. On the other hand,

which shows that .

We have examined several functions in order to determine their inverse functions, but there is still more to this story. Not every function has an inverse function, so we must learn how to check for this situation.

**not**have an inverse function. Imagine that it did. Then and . Then we have both and . Since a

**function**cannot send the same input to two different outputs, must not have an inverse function.

Look again at the last question. If two different inputs for a function have the same output, there is no hope of that function having an inverse function. Why? This is because the inverse function must also be a function, and a function can only have one output for each input. More specifically, we have the next definition.

- Since words may have more than one definition, “relating words to their definition in a dictionary” is not a function.
- Since every social security number corresponds exactly to one person, “relating social security numbers of living people to actual living people” is a function. Also, since each person has exactly one social security number, it is one-to-one.
- Since every person only has one birth date, “relating people to their birth date” is a function. However, many people have the same birth date, hence this function is not one-to-one.
- Since mothers can have more (or less) than one child, “relating mothers to their children” is not a function.

You may recall that a plot gives as a function of if every vertical line crosses the
plot at most once, and we called this the **vertical line test**. Similarly, a function is
one-to-one if every horizontal line crosses the plot at most once, and we call this the
**horizontal line test**.

**horizontal line test**.

As we have discussed, we can only find an inverse of a function when it is one-to-one. If a function is not one-to-one, but we still want an inverse, we must restrict the domain. Let’s see what this means in our next examples.

This idea of restricting the domain is critical for understanding functions like .

**square-root is a function**, we must have only one output value when we plug in . We choose the positive square-root, meaning that