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: