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This module describes what inverse functions are and how they can be used.

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 ( t ) = 100 t .

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 ( m ) = 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:

m ( t ) = 100 t size 12{m \( t \) ="100"t} {}
t ( m ) = m / 100 size 12{t \( m \) =m/"100"} {}

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

3 m ( t ) = 100 t 300 size 12{3 rightarrow m \( t \) ="100"t rightarrow "300"} {}
300 t ( m ) = m / 100 3 size 12{"300" rightarrow t \( m \) =m/"100" rightarrow 3} {}
Inverse functions

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:

f ( x ) = 3x + 7 size 12{f \( x \) =3x+7} {}
g ( x ) = 1 3 x 7 size 12{g \( x \) = { { size 8{1} } over { size 8{3} } } x - 7} {}

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

2 3x + 7 13 size 12{2 rightarrow 3x+7 rightarrow "13"} {}
13 3 x - 7 13 3 - 7 - 8 3 size 12{"13" rightarrow 1/3x-7 rightarrow "13"/3-7 rightarrow -8/3 } {}
Not inverse functions

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:

f ( x ) = 3x + 7 size 12{f \( x \) =3x+7} {}
g ( x ) = 1 3 x 7 size 12{g \( x \) = { { size 8{1} } over { size 8{3} } } left (x - 7 right )} {}

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

2 3x + 7 13 size 12{2 rightarrow 3x+7 rightarrow "13"} {}
13 1 3 x 7 2 size 12{"13" rightarrow { { size 8{1} } over { size 8{3} } } left (x - 7 right ) rightarrow 2} {}
Inverse 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:

F ( C ) = 9 5 C + 32 size 12{F \( C \) = left ( { {9} over {5} } right )C+"32"} {}
C ( F ) = 5 9 F 32 size 12{C \( F \) = left ( { {5} over {9} } right ) left (F - "32" right )} {}

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

The notation and definition of an inverse function

The notation for the inverse function of f ( x ) size 12{f \( x \) } {} is 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 ) size 12{f rSup { size 8{ - 1} } \( x \) } {} simply means “the inverse function of f ( x ) size 12{f \( x \) } {} .” It is defined formally by the fact that if you plug any number x 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 ) ) = x size 12{f \( f rSup { size 8{ - 1} } \( x \) \) =x} {} .

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Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
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