1.12 Inverse functions

Let's go back to Alice, who makes $100/day. We know how to answer questions such as "After 3 days, how much money has she made?" We use the function $m\left(t\right)=100t$ .

But suppose I want to ask the reverse question: “If Alice has made $300, how many hours has she worked?” This is the job of an inverse function. It gives the same relationship, but reverses the dependent and independent variables. $t\left(m\right)=m/100$ . Given any amount of money, divide it by 100 to find how many days she has worked.

• If a function answers the question: “Alice worked this long, how much money has she made?” then its inverse answers the question: “Alice made this much money, how long did she work?"
• If a function answers the question: “I have this many spoons, how much do they weigh?” then its inverse answers the question: “My spoons weigh this much, how many do I have?”
• If a function answers the question: “How many hours of music fit on 12 CDs?” then its inverse answers the question: “How many CDs do you need for 3 hours of music?”

How do you recognize an inverse function?

Let’s look at the two functions above:

Mathematically, you can recognize these as inverse functions because they reverse the inputs and the outputs .

Of course, this makes logical sense. The first line above says that “If Alice works 3 hours, she makes $300.” The second line says “If Alice made $300, she worked 3 hours.” It’s the same statement, made in two different ways.

But this “reversal” property gives us a way to test any two functions to see if they are inverses. For instance, consider the two functions:

They look like inverses, don’t they? But let’s test and find out.

The first function turns a 2 into a 13. But the second function does not turn 13 into 2. So these are not inverses.

On the other hand, consider:

Let’s run our test of inverses on these two functions.

So we can see that these functions do, in fact, reverse each other: they are inverses.

A common example is the Celsius-to-Fahrenheit conversion:

where $C$ is the Celsius temperature and $F$ the Fahrenheit. If you plug $\text{100}°C$ into the first equation, you find that it is $\text{212}°F$ . If you ask the second equation about $\text{212}°F$ , it of course converts that back into $\text{100}°C$ .

The notation and definition of an inverse function

The notation for the inverse function of $f(x)$ is $f^{-1}(x)$ . This notation can cause considerable confusion, because it looks like an exponent, but it isn’t. $f^{-1}(x)$ simply means “the inverse function of $f(x)$ .” It is defined formally by the fact that if you plug any number $x$ into one function, and then plug the result into the other function, you get back where you started. (Take a moment to convince yourself that this is the same definition I gave above more informally.) We can represent this as a composition function by saying that $f(f^{-1}(x))=x$ .