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