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 CC size 12{C} {} is the Celsius temperature and FF size 12{F} {} the Fahrenheit. If you plug 100°C100°C size 12{"100"°C} {} into the first equation, you find that it is 212°F212°F size 12{"212"°F} {}. If you ask the second equation about 212°F212°F size 12{"212"°F} {}, it of course converts that back into 100°C100°C size 12{"100"°C} {}.

The notation for the inverse function of f(x)f(x) size 12{f \( x \) } {} is f−1(x)f−1(x) size 12{f rSup { size 8{ - 1} } \( x \) } {}. This notation can cause considerable confusion, because it looks like an exponent, but it isn’t. f−1(x)f−1(x) size 12{f rSup { size 8{ - 1} } \( x \) } {} simply means “the inverse function of f(x)f(x) size 12{f \( x \) } {}.” It is defined formally by the fact that if you plug any number xx size 12{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))=xf(f−1(x))=x size 12{f \( f rSup { size 8{ - 1} } \( x \) \) =x} {}.

Definition 1: Inverse Function

f−1(x)f−1(x) size 12{f rSup { size 8{ - 1} } \( x \) } {} is defined as the inverse function of f(x)f(x) size 12{f \( x \) } {} if it consistently reverses the f(x)f(x) size 12{f \( x \) } {} process. That is, if f(x)f(x) size 12{f \( x \) } {} turns aa size 12{a} {} into bb size 12{b} {}, then f−1(x)f−1(x) size 12{f rSup { size 8{ - 1} } \( x \) } {} must turn bb size 12{b} {} into aa size 12{a} {}. More concisely and formally, f−1(x)f−1(x) size 12{f rSup { size 8{ - 1} } \( x \) } {} is the inverse function of f(x)f(x) size 12{f \( x \) } {} if f(f−1(x))=xf(f−1(x))=x size 12{f \( f rSup { size 8{ - 1} } \( x \) \) =x} {}.

In examples above, we saw that if f(x)=3x+7f(x)=3x+7 size 12{f \( x \) =3x+7} {}, then f−1(x)=13x−7f−1(x)=13x−7 size 12{f rSup { size 8{ - 1} } \( x \) = { { size 8{1} } over { size 8{3} } } left (x - 7 right )} {}. We also saw that the function 13x−713x−7 size 12{ { { size 8{1} } over { size 8{3} } } x - 7} {}, which may have looked just as likely, did not work as an inverse function. So in general, given a function, how do you find its inverse function?

Remember that an inverse function reverses the inputs and outputs. When we graph functions, we always represent the incoming number as xx size 12{x} {} and the outgoing number as yy size 12{y} {}. So to find the inverse function, switch the x and y values, and then solve for yy size 12{y} {}.

Were you surprised by the answer? At first glance, it seems that the numbers in the original function (the 2, 3, and 5) have been rearranged almost at random.

But with more thought, the solution becomes very intuitive. The original function f(x)f(x) size 12{f \( x \) } {} described the following process: double a number, then subtract 3, then divide by 5. To reverse this process, we need to reverse each step in order: multiply by 5, then add 3, then divide by 2. This is just what the inverse function does.

Some functions have no inverse function. The reason is the rule of consistency.

For instance, consider the function y=x2y=x2 size 12{y=x rSup { size 8{2} } } {}. This function takes both 3 and –3 and turns them into 9. No problem: a function is allowed to turn different inputs into the same output. However, what does that say about the inverse of this particular function? In order to fulfill the requirement of an inverse function, it would have to take 9, and turn it into both 3 and –3—which is the one and only thing that functions are not allowed to do. Hence, the inverse of this function would not be a function at all!

Table 4: If 3 goes in, 9 comes out. If –3 goes in, 9 also comes out. No problem:

3 → − 3 → 3 → − 3 →

9 → 9 → 9 → 9 →

Table 5: But its inverse would have to turn 9 into both 3 and –3. No function can do this, so there is no inverse.

9 → 9 → 9 → 9 →

3 → − 3 → 3 → − 3 →

In general, any function that turns multiple inputs into the same output, does not have an inverse function.

What does that mean in the real world? If we can convert Fahrenheit to Celsius, we must be able to convert Celsius to Fahrenheit. If we can ask “How much money did Alice make in 3 days?” we must surely be able to ask “How long did it take Alice to make $500?” When would you have a function that cannot be inverted?

Let’s go back to this example:

Recall the example that was used earlier: “Max threw a ball. The height of the ball depends on how many seconds it has been in the air.” The two variables here are hh size 12{h} {} (the height of the ball) and tt size 12{t} {} (the number of seconds it has been in the air). The function h(t)h(t) size 12{h \( t \) } {} enables us to answer questions such as “After 3 seconds, where is the ball?”

The inverse question would be “At what time was the ball 10 feet in the air?” The problem with that question is, it may well have two answers!

So what does that mean? Does it mean we can’t ask that question? Of course not. We can ask that question, and we can expect to mathematically find the answer, or answers—and we will do so in the quadratic chapter. However, it does mean that time is not a function of height because such a “function” would not be consistent: one question would produce multiple answers.