# Find the range of each of the following functions.

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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

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(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