THE INVERSE of a function undoes the action of that function.

Say, for example, that a function  f  acts on 5, producing  f(5).  Then if g is the inverse of f, then g acting on f(5) will bring back 5

g(f(5)) = 5.

Actually, g must do that for all values in the domain of f.  And f must do that for all values in the domain of g.  Here is the definition:

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

(If we take the graph of the left to be the right-hand branch of y = x², then the graph on the right is its inverse, y = .)

To see that in fact that is the graph of the inverse, let A be any point on

the graph of f(x), let its coordinates be (a, b), let it be a distance d from the origin C, and let AC make 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 of D  90° counterclockwise to the point C'.

To see that the coordinates of C' are (b, a), consider that since angle C'A'D is 90°, then C'A' makes an angle of  90° − θ  with the x-axis.  That is, angle C'A'B' is the complement of angle B'A'D, which is angle θ. Therefore in the right triangle A'B'C', the angle at C' is equal to θ.