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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

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Solution

(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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