Consider the function f: R → R, defined by
Write its domain and range. Also, draw the graph of f(x).
Given:
f(x)
To Find:
Domain and Range of f(x)
When f(x) = 1 – x | x < 0
In this case there is no value of x (x < 0) which makes the above expression undefined.
Therefore,
Domain(f) = (-∞, 0) …(1)
When f(x) = x | x = 0
In this case there is no value other than 0 which makes the above expression undefined.
Therefore,
Domain(f) = 0 …(2)
When f(x) = x + 1 | x > 0
In this case there is no value of x (x > 0) which makes the above expression undefined.
Therefore,
Domain(f) = (0, ∞) …(3)
From equations (1),(2) & (3) We can say that the domain of f(x) as a whole :
Domain(f) = (-∞, ∞)
Now when, f(x) = 1 -x
x = 1 – f(x)
As x ranges from -∞ to 0, then f(x) ranges from 1 to ∞
Therefore,
Range(f) = (1, ∞) …(4)
Now when, f(x) =x
As x = 0
Therefore,
Range(f) = 0 …(5)
Now when, f(x) = x +1
x = f(x) - 1
As x ranges from 0 to ∞, then f(x) ranges from 1 to ∞
Therefore,
Range(f) = (1, ∞) …(6)
From (4), (5) & (6) the range of f(x) as whole:
Range(f) = 0 υ (1, ∞)
Graph: