We are familiar with the least value, greatest value and range of the most standard functions of all origin. Consider constant, identity, reciprocal, modulus, greatest integer, least integer, fraction part, trigonometric, inverse trigonometric, exponential and logarithmic functions. All these functions have been described in detail and we know their properties with respect to least and greatest values and also the range. Greatest value of sine function, for example, is 1. On the other hand, exponential and logarithmic functions etc. neither have minimum (therefore least value) nor maximum (therefore greatest value). However, these functions have least and greatest in finite interval in accordance with mean value theorem.

In case, the function can be reduced to the standard forms having least and greatest values, then it is possible to know its range. In the example, we consider one such trigonometric function.

### Example 1

Problem : Find the range of the continuous function given by :

where “a” and “b” are constants.

Solution : Here, given function is addition of two trigonometric functions. As we know least and greatest values of sine and cosine functions, we shall attempt to reduce given function in terms of either sine or cosine function (note that the algorithm for reducing addition of sine and cosine functions as presented here is a standard algorithm. We should also note that this algorithm, as a matter of fact, is used in analyzing superposition principle of waves) :

where,

Substituting in the given function, we have :

We know that minimum and maximum values of cosine function are "-1" and "1" respectively. Hence,

Therefore, range of the given function is :