## Repetition, Square Root

The square root of a non-negative number  is a non-negative number whose square is equal to . This number is denoted by ; the number  is called the sub-root.

Examples:

1.  (, )

2.  (, )

Note:

, but  – the root cannot be equal to a negative number.

– cannot be calculated. The square root of a negative number does not exist.

## Function

A function , where , is a law that maps a number  to each non-negative number .

We already know the function . It is a function of type . So we will study the function  on the basis of the function  where .

## The graph of a function

The graph of function  is a branch of a parabola. Let us check this by making a table.

 x 0 1 4 6.25 9 0 1 2 2.5 3

Draw the found points on the coordinate plane (see Fig. 1).

Fig. 1. The graph of the function

If the argument increases from 0 to , the function increases from 0 to .

## Properties of the function

1. The set of values of the function  is ray .

Let us prove this property

The proof is

Let  be an arbitrary number from the interval . Can we find a number  such that ? To find it, solve the equation:

The number  is reached when the argument is equal to  (see Figure 2).

Figure 2. Illustration of the proof

Hence:

This was required to prove it.

Consequences of this property

a) The function  is not bounded from above. That is, there is no largest positive number on the Y axis for this function.

b) The function is bounded from below and has the smallest value.

c)  for all .

2. The function  increases monotonically over the entire range of determination, that is, at .

Note:

A function is called monotonically increasing over the entire domain of determination if, for any  and  belonging to the domain of determination, the inequality  follows from the inequality .

Figure 3 illustrates to us that the function  is monotonically increasing.

Fig. 3. Function  is monotonically increasing

3. The function  is convex upwards on the entire domain of definition.

For any two points, such as  and  (see Figure 4), the arc lying between these points will be over the segment connecting these two points, hence the function is convex upwards.

Figure 4. Function  convex upwards