## 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.
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.