Question

Find the range of each of the following functions.
(i) f(x) = 2 – 3x, x ∈ R, x > 0.
(ii) f(x) = x² + 2, x, is a real number.
(iii) f(x) = x, x is a real number

Answer 1

(i) f(x) = 2 – 3x, x ∈ R, x > 0
The values of f(x) for various values of real numbers x > 0 can be written in the tabular form as

Thus, it can be clearly observed that the range of f is the set of all real
numbers less than 2.
i.e., range of f = (– ∞, 2)

Alter:
Let x > 0
⇒ 3x > 0
⇒ 2 –3x < 2
⇒ f(x) < 2
∴Range of f = (– ∞, 2)
(ii) f(x) = x² + 2, x, is a real number
The values of f(x) for various values of real numbers x can be written in the tabular form as

Thus, it can be clearly observed that the range of f is the set of all real
numbers greater than 2.
i.e., range of f = [2,∞)
Alter:
Let x be any real
number. Accordingly,
x² ≥ 0
⇒ x² + 2 ≥ 0 + 2
⇒ x² + 2 ≥ 2
⇒ f(x) ≥ 2
∴ Range of f = [2,∞)
(iii) f(x) = x, x is a real number
It is clear that the range of f is the set of all real
numbers. ∴ Range of f = R