Question 5:

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

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= (−∞, 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

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