Power function with rational exponent. Function

Recall the properties and graphs of power functions with a negative integer exponent.

For even n, :

Function example:

All graphs of such functions pass through two fixed points: (1;1), (-1;1). A feature of functions of this type is their parity, the graphs are symmetrical with respect to the op-y axis.

Rice. 1. Graph of a function

For odd n, :

Function example:

All graphs of such functions pass through two fixed points: (1;1), (-1;-1). A feature of functions of this type is their oddness, the graphs are symmetrical with respect to the origin.

Rice. 2. Function Graph

Let us recall the main definition.

The degree of a non-negative number a with a rational positive exponent is called a number.

The degree of a positive number a with a rational negative exponent is called a number.

For the following equality holds:

For example: ; - the expression does not exist by definition of a degree with a negative rational exponent; exists, since the exponent is an integer,

Let us turn to the consideration of power functions with a rational negative exponent.

For example:

To plot this function, you can make a table. We will do otherwise: first, we will build and study the graph of the denominator - we know it (Figure 3).

Rice. 3. Graph of a function

The graph of the denominator function passes through a fixed point (1;1). When constructing a graph of the original function, this point remains, when the root also tends to zero, the function tends to infinity. And, conversely, as x tends to infinity, the function tends to zero (Figure 4).

Rice. 4. Function Graph

Consider one more function from the family of functions under study.

It is important that by definition

Consider the graph of the function in the denominator: , we know the graph of this function, it increases in its domain of definition and passes through the point (1; 1) (Figure 5).

Rice. 5. Function Graph

When constructing a graph of the original function, the point (1; 1) remains, when the root also tends to zero, the function tends to infinity. And, conversely, as x tends to infinity, the function tends to zero (Figure 6).

Rice. 6. Function Graph

The considered examples help to understand how the graph goes and what are the properties of the function under study - a function with a negative rational exponent.

Graphs of functions of this family pass through the point (1;1), the function decreases over the entire domain of definition.

Function scope:

The function is not bounded from above, but bounded from below. The function has neither a maximum nor a minimum value.

The function is continuous, it takes all positive values ​​from zero to plus infinity.

Convex Down Function (Figure 15.7)

Points A and B are taken on the curve, a segment is drawn through them, the entire curve is below the segment, this condition is satisfied for arbitrary two points on the curve, therefore the function is convex downward. Rice. 7.

Rice. 7. Convexity of a function

It is important to understand that the functions of this family are bounded from below by zero, but they do not have the smallest value.

Example 1 - find the maximum and minimum of the function on the interval and increases on the interval )