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

x	0.01	0.1	0.9	1	2	2.5	4	5	...
f(x)	1.97	1.7	-0.7	-1	-4	-5.5	-10	-13	...

Thus, it can be clearly observed that the range of fis 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 xcan be written in the tabular form as

x	0	±0.3	±0.8	±1	±2	±3		..…
f(x)	2	2.09	2.64	3	6	11		..…

Thus, it can be clearly observed that the range of fis 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 fis the set of all real numbers.
∴ Range of f = R