Inverse Functions

An inverse function goes in the opposite direction!

Let us start with an example:

Here we have the function f(x) = 2x+3, written as a flow diagram:

2x+3

The Inverse Function just goes the other way:

Inverse

So the inverse of:   2x+3   is:   (y-3)/2

 

The inverse is usually shown by putting a little "-1" after the function name, like this:

f-1(y)

We say "f inverse of y"

So, the inverse of f(x) = 2x+3 is written:

f-1(y) = (y-3)/2

(I also used y instead of x to show that we are using a different value.)

Back to Where We Started

The cool thing about the inverse is that it should give us back the original value:


When the function f turns the apple into a banana,
Then the inverse function f-1 turns the banana back to the apple


Example:

Using the formulas from above, we can start with x=4:

f(4) = 2×4+3 = 11

We can then use the inverse on the 11:

f-1(11) = (11-3)/2 = 4

And we magically get 4 back again!

We can write that in one line:

f-1( f(4) ) = 4

"f inverse of   f of 4   equals 4"

So applying a function f and then its inverse f-1 gives us the original value back again:

f-1( f(x) ) = x

We could also have put the functions in the other order and it still works:

f( f-1(x) ) = x

Example:

Start with:

f-1(11) = (11-3)/2 = 4

And then:

f(4) = 2×4+3 = 11

So we can say:

f( f-1(11) ) = 11

"f of   f inverse of 11   equals 11"

Solve Using Algebra

We can work out the inverse using Algebra. Put "y" for "f(x)" and solve for x:

The function:   f(x) = 2x+3
Put "y" for "f(x)":   y = 2x+3
Subtract 3 from both sides:   y-3 = 2x
Divide both sides by 2:   (y-3)/2 = x
Swap sides:   x = (y-3)/2
         
Solution (put "f-1(y)" for "x") :   f-1(y) = (y-3)/2

This method works well for more difficult inverses.

Fahrenheit to Celsius

A useful example is converting between Fahrenheit and Celsius:

To convert Fahrenheit to Celsius:   f(F) = (F - 32) x 5/9
The Inverse Function (Celsius back to Fahrenheit) is:   f-1(C) = (C × 9/5) + 32

For You: see if you can do the steps to create that inverse!

Inverses of Common Functions

It has been easy so far, because we know the inverse of Multiply is Divide, and the inverse of Add is Subtract, but what about other functions?

Here is a list to help you:

Inverses Careful!
add <=> Subtract  
Multiply <=> Divide Don't divide by zero
1/x <=> 1/y x and y not zero
x^2 <=> sqrt-y x and y ≥ 0
x^n <=> nth-root-y or y^(1/n) n not zero
(different rules when n is odd, even, negative or positive)
e^x <=> ln(y) y > 0
a^x <=> log_a(y) y and a > 0
sin(x) <=> sin-1(y) -π/2 to +π/2
cos(x) <=> cos-1(y) 0 to π
tan(x) <=> tan-1(y) -π/2 to +π/2

(Note: you can read more about Inverse Sine, Cosine and Tangent.)

Careful!

Did you see the "Careful!" column above? That is because some inverses work only with certain values.

Example: Square and Square Root

When we square a negative number, and then do the inverse, this happens:

Square:   (-2)2 = 4
Inverse (Square Root):   √(4) = 2

 

But we didn't get the original value back! We got 2 instead of -2. Our fault for not being careful!

So the square function (as it stands) does not have an inverse

But we can fix that!

Restrict the Domain (the values that can go into a function).

Example: (continued)

Just make sure we don't use negative numbers.

In other words, restrict it to x ≥ 0 and then we can have an inverse.

So we have this situation:

  • x2 does not have an inverse
  • but {x2 | x ≥ 0 } (which says "x squared such that x is greater than or equal to zero" using set-builder notation) does have an inverse.

Why Would There Be No Inverse?

Let us see graphically what is going on here:

To be able to have an inverse we need unique values.

Just think ... if there are two or more x-values for one y-value, how do we know which one to choose when going back?.

noNo Inverse   yesInverse is Possible
General Function   Injective Function
When a y-value has more than one x-value, how do we know which x-value to go back to?   When there is a unique y-value for every x-value we can always "go back" from y to x.

So we have this idea of "a unique y-value for every x-value", and it actually has a name. It is called "Injective" or "One-to-one":

When a function is "One-to-one" (Injective) it has an inverse.

Domain and Range

So what is all this talk about "Restricting the Domain"?

doman and range

In its simplest form the domain is all the values that go into a function (and the range is all the values that come out).

As it stands the function above does not have an inverse.

But we could restrict the domain so there is a unique y for every x ... doman and range
... and now we can have an inverse:

 

Note also:

  • The function f(x) goes from the domain to the range,
  • The inverse function f-1(y) goes from the range back to the domain.
doman and range

Or...

We could plot them both in terms of x ... so it is now f-1(x), not f-1(y).

doman and range

f(x) and f-1(x) are like mirror images
(flipped about the diagonal).

In other words:

The graph of f(x) and f-1(x) are symmetric across the line y=x

 

Example: Square and Square Root (continued)

First, we restrict the Domain to x ≥ 0:

  • {x2 | x ≥ 0 } "x squared such that x is greater than or equal to zero"
  • {√x | x ≥ 0 } "square root of x such that x is greater than or equal to zero"

 

x^2 vs square  root x
And you can see they are "mirror images"
of each other about the diagonal y=x.

 

Note: we could have restricted the domain to x ≤ 0 and the inverse would then be f-1(x) = -√x:

  • {x2 | x ≤ 0 }
  • {-√x | x ≥ 0 }

Which are inverses, too.

  x^2 vs -square  root x

 

Not Always Solvable!

It is sometimes not possible to find an Inverse of a Function.

Example: f(x) = x/2 + sin(x)

We cannot work out the inverse of this, because we cannot solve for "x":

y = x/2 + sin(x)

y ... ? = x

Notes on Notation

Even though we write f-1(x), the "-1" is not an exponent (or power):

f-1(x) ...is different to... f(x)-1
Inverse of the function f   f(x)-1 = 1/f(x)
(the Reciprocal)

Summary

  • The inverse of f(x) is f-1(y)
  • We can find an inverse by reversing the "flow diagram"
  • Or we can find an inverse by using Algebra:
    • Put "y" for "f(x)", and
    • Solve for x
  • We may need to restrict the domain for the function to have an inverse