# CLASS-11RELATION & FUNCTION - REAL VALUE FUNCTION

Real Value Functions

Real Value Function -

If R be the set of real numbers and A, B are subsets of R, then the function f : A → B is called a real function or real valued function.

Piece Function

Although functions are frequently described by single formulae, it is not necessary that it should always be so. Thus sometimes a function is defined in two or more parts, such as the function ‘g’ defined by  -

g(x) = x, for each number x such that x ≥ 0,

g(x) = - x, for each number x such that x < 0.

This is a single rule as it defines one function, even though it involves two equations. It is customary

x ;  x ≥ 0

to abbreviate this rule in the form g(x) =  {

- x ;  x < 0

The domain of this function is the set of all real numbers and the range is the set of all non-negative real numbers. The above function is just the absolute value function g(x) = ǀ x ǀ, x ∈ R.

Suppose we have a piece function ‘h’ defined as

- 1, x < 0

h(x) = {  0, x = 0

1, x > 0

We notice that ‘h’ assigns a number to every real number, hence the domain of ‘h’ is the set of all real numbers. Now, since h (every negative real number) = - 1, h(0) = 0, h (every positive real number) = 1, therefore, the range of the function h defined as above, is the set {- 1, 0, 1}

2x–1,when x ≤ 0                     1           -1

Example.1) If f(x) = {                       , then find f(-----)and f(----)

x², when  x > 0                      2            2

Ans.)

When x ≤ 0, f(x) = 2x – 1

- 1          - 1

f(-----) = 2 (-----) – 1 =  - 2

2             2

When x > 0, f(x) = x²

1          1           1

So, f(-----) = (-----)² = -----     (Ans.)

2          2           4