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