Other Functions

Identity function

A function f: A→A is said to be an identity function on A if f(x) = x for all x ∈A. That is, every element of A is mapped onto itself. An identity function is a one–one and onto function. Its inverse is itself.

Constant function

A function f: A→B is called a constant function if there is an element c ∈B such that f(x) = c for all x ∈A.

Example 1:

Let f: R→R be defined by f(x) = 2 then f is a constant function.

Equal function

The functions f and g having the same domain A are said to be equal if f(x) = g(x) for all x ∈A. This is written as f = g.

Example 2:

Let f(x) = x - 2    and

     g(x) = x2 – 4
               x + 2 

x ≠ 2  where    f: R-{-2} →R      g: R-{-2} →R.

Show that f(x) = g(x).

f(x)  = x - 2

g(x) = x2 - 4
          x + 2 

       = (x - 2) (x +2)
               (x + 2) 

       = x - 2             since x ≠ -2

f(x) = g(x) for all x ∈R - {-2}.

So  f = g

Try these questions

  1. Determine whether f(x) and g(x) are identical functions.
    Two function forms are equivalent as f(x) is reduced to g(x) on simplification. Now, expression of f(x) is defined for all values of x except x=0. Thus, domain of f(x) is R-{0}. On the other hand, domain of reciprocal function g(x) is also R-{0}. Clearly, two given functions are equal.

  2. Which one of the following equations represents a constant function?
    1. x = 3
    2. 2x + y = 4
    3. y = 6
    4. 2y + x = 12
    Answer: C
    A constant function is of the form y = b, where b is a constant. So, among the equations listed, only y = 6 represents a constant function.