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.
   f(x)=xx2
   g(x)=1x
   Answer:
   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.