Functions f(x) and g(x) are inverses of one another if:

f(g(x)) = x   and   g(f(x)) = x,

for all values in their respective domains.

Example 1.   Let f(x) = x + 2,   and   g(x) = x − 2.  Then they are inverses of one another.  For g(x), which subtracts 2 from a number, is the inverse of adding 2:  f(x).

Constructing the inverse

When we have a function y = f(x) -- for example

y = x²

-- then we can often "invert" the equation by solving for x.  In this case,

x now appears as a function of y.  Therefore on exchanging the variables,

is the inverse function of  y = x².

(Taking the square root of a number is the inverse of squaring a number.)

Hence, to construct the inverse of a function y = f(x):

Solve for x, then exchange the variables.

Example 2.   What function is the inverse of  y = 3x + 4?

 Solution.   Exchange the sides of the equation, and solve for x:

That function is the inverse of  y = 3x + 4.

Problem 2.   What function is the inverse of  y = 5x?

On solving for x:

Clearly, dividing by 5 is the inverse of multiplying by 5.

Problem 3.   a)   Let y = f(x) = x − 4.  Construct its inverse, g(x).

b)   Prove that f(x) and g(x) are inverses.

f(g(x)) = f(x + 4) = (x + 4) − 4 = x,

and

g(f(x)) = g(x − 4) = (x − 4) + 4 = x.

Notation

The function  I(x) = x  is called the identity function.  It always returns x.

As a notation for the inverse of a function f, we sometimes see  f ⁻¹  ("f inverse").  "−1" is not an exponent.  That notation is used because in the language of composition of functions, we can write:

f o f ⁻¹ = I

This is similar in form to the multiplication of numbers,  a· a⁻¹ = 1.

The graph of an inverse function

The graph of the inverse of a function f(x) can be found as follows:

Reflect the graph about the x-axis, then rotate it 90° counterclockwise.

To see that in fact that is the graph of the inverse, consider the following:

Let A be a point on the graph of f(x) whose coordinates are (x, y), and which is a distance d from the origin C;  AC makes an angle θ with the x-axis; triangle ABC is right angled.

The figure on the left shows the reflection of A about the x-axis to the point D.  The figure on the right shows the rotation 90° counterclockwise to the point C'.  In that figure, C'A' makes an angle of  90° − θ  with the x-axis.  That is, in the right triangle A'B'C', angle C'A'B' is the complement of angle θ.  Therefore the angle at C' is equal to θ.

In other words, when each point (x, y) on f(x) is transformed into (y, x), then a graph of a functions g(x) results.  And we claim that the graph of
g(x) is the graph of the inverse of f(x).

For, when g operates on its x-coordinate -- which is the y-coordinate of  f -- it produces x:

g( y ) = x.

But y is f(x):

g( f(x)) = x.