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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
Asked by Yashraj Last Modified
Answer (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 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
| 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 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
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