Sum

Find the range of each of the following functions.

*f*(*x*) = *x*^{2} + 2, *x*, is a real number.

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#### Solution

*f*(*x*) = *x*^{2} + 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,`oo`)

**Alter:**

Let *x* be any real number.

Accordingly,

*x*^{2} ≥ 0

⇒ *x*^{2} + 2 ≥ 0 + 2

⇒ *x*^{2} + 2 ≥ 2

⇒ *f*(*x*) ≥ 2

∴ Range of *f* = [2, `oo`)

Concept: Some Functions and Their Graphs

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