Power functions and their graphics. Indicative function - properties, graphs, formulas

). With valid base values h. and indicator but Usually consider only valid values \u200b\u200bof S. F. x a. They exist, in any case, for all x\u003e 0; if a but -rational number with an odd denominator, then they also exist for everyone x 0; if the denominator of the rational number but even if irrationally, then x A. does not have a valid value x 0 x \u003d0 Power Function x A. equal to zero for all but \u003e 0 and not defined when a 0; 0 ° has no meaning. S. F. (in the field of valid values) is unambiguous, except in cases where but - The rational number depicted by an incomprehensible fraction with an even denominator: in these cases it is double-digit, and its values \u200b\u200bfor the same value of the argument h.\u003e 0 are equal in absolute value, but are opposed to the sign. Typically, only non-negative, or arithmetic, value S. F. For h.> 0 S. F. - increasing if but\u003e 0, and descending, if butx \u003d 0, in the case of 0 ax A.)"\u003d AX A-1.Further,

Functions of type y \u003d cx a, Where from - permanent coefficient, play an important role in mathematics and its applications; for but \u003d 1 These functions express direct proportionality (their graphics are direct passing through the origin of the coordinates, see fig. one), for a \u003d.-1 - inverse proportionality (graphics - equilateral hyperboles with the center at the beginning of the coordinates having the axis of coordinate with their asymptotes, see fig. 2.). Many laws of physics are mathematically expressed using the functions of the type y \u003d cx a(see fig. 3.); eg, y \u003d CX 2 expresses the law of an equidal or equible movement ( y - way, x - Time, 2. c. - acceleration; The initial path and speed are zero).

In the complex region S. F. z. A is determined for all z. ≠ 0 Formula:

where k.\u003d 0, ± 1, ± 2, .... if but - whole, then S. F. z. A unambiguous:

If a but - rational (A. \u003d P / Q, Where rand q. Mutually simple), then S. F. z A. Accept q. Different values:

where ε k \u003d - Degree roots q. From one: k \u003d 0, 1, ..., q - 1. If but - irrational, then S. F. z. A - infinified: multiplier ε α2κ. π ι Takes for different k. Various values. With complex values \u200b\u200band S. F. z A. Determined by the same formula (*). For example,

so, in particular, K \u003d 0, ± 1, ± 2, ....

Under the main meaning ( z A.) 0 S. F. It is understood by its meaning when k \u003d. 0, if -πz ≤ π (or 0 ≤ arg z. z a) = |z A.|e Ia Arg Z, (i.) 0 \u003d E -π / 2, etc.

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

Watch what is a "power function" in other dictionaries:

The function of the form y \u003d axn, where a and n any actual numbers ... Big Encyclopedic Dictionary

The power function function, where (indicator of the degree) Some real number ... Wikipedia

The form of the form y \u003d ahn, where a and n act. Numbers, S. F. It covers a large number of patterns in nature. In fig. depicted graphics S. F. For n \u003d 1, 2, 3, 1/2 and a \u003d 1. To art. Power function … Big Encyclopedic Polytechnic Dictionary

The function of the form y \u003d axn, where a and n any actual numbers. The figure shows the graphs of the power function for n \u003d 1, 2, 3, 1/2 and a \u003d 1. * * * The power function is a power function, the function of the form y \u003d AXN, where a and n any actual numbers ... encyclopedic Dictionary

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Function y \u003d x A, where and a constant number. If a integer, then S. F. Private case of rational function. With complex values \u200b\u200bof hi ac. f. ambiguous, if a non-tariff. With fixed valid. And and the number X is the degree ... Mathematical encyclopedia

The function of the form y \u003d ahn, where a and n any valid numbers. In fig. depicted graphics S. F. For n \u003d 1, 2, 3, 1/2 and a \u003d 1 ... Natural science. encyclopedic Dictionary

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Demand function - function reflecting the dependence of the amount of demand for individual goods and services (consumer goods) from the complex of factors affecting it. Thin narrow interpretation: F.S.Ch. examines interdependence between demand for goods and price ... ... Economics and Mathematical Dictionary

Y \u003d 1 + x + x2 + x3 + ... is defined for real or complex values \u200b\u200bof x, moduliotors less than one. F. Species y \u003d p0xn + p1xn 1 + p2xn 2 + ... + pn 1x + pn, where coefficients, p0, p1, p2, ..., pn data of the number is called. The bottom function N oh ... ... Encyclopedia Brockhaus and Ephron

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Recall the properties and graphs of power functions with a whole negative indicator.

With even N,:

Example function:

All graphs of such functions pass through two fixed points: (1; 1), (-1; 1). The feature of the functions of this species is their parity, graphics are symmetrical relative to the OU axis.

Fig. 1. Function schedule

With odd N,:

Example function:

All graphs of such functions pass through two fixed points: (1; 1), (-1; -1). The feature of the functions of this species is their oddness, graphics are symmetrical relative to the start of coordinates.

Fig. 2. Schedule function

Recall the basic definition.

The degree of nonnegative number and a rational positive indicator is the number.

The degree of positive number and with a rational negative indicator is called the number.

Equality is performed:

For example: ; - the expression does not exist to determine the degree with a negative rational indicator; There is, since the indicator is a whole,

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

For example:

To build a graph of this feature, you can create a table. We will proceed otherwise: first we will build and study the schedule of the denominator - it is known for us (Figure 3).

Fig. 3. Function graph

The graph of the function of the denominator passes through a fixed point (1; 1). When constructing a graph of the source function, this point remains, with the root also tends to zero, the function tends to infinity. And, on the contrary, with the desire of x to infinity, the function tends to zero (Figure 4).

Fig. 4. Function schedule

Consider another feature from the family of studied functions.

It is important that by definition

Consider the schedule of the function in the denominator :, the schedule of this function is known to us, it increases on its definition area and passes through the point (1; 1) (Figure 5).

Fig. 5. Function schedule

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, on the contrary, with the desire of x to infinity, the function tends to zero (Figure 6).

Fig. 6. Function graph

Considered examples help to understand how the schedule passes and what the properties of the function being studied are functions with a negative rational indicator.

The graphs of the functions of this family pass through the point (1; 1), the function decreases throughout the definition area.

Function definition area:

The function is not limited from above, but is limited to below. The function does not have the greatest nor the smallest value.

The function is continuous, takes all the positive values \u200b\u200bfrom zero to the plus infinity.

Function convex down (Figure 15.7)

Points A and B were taken on the curve, through them a segment was taken, the whole curve is below the segment, this condition is performed for arbitrary two points on the curve, therefore, the function is convexing down. Fig. 7.

Fig. 7. Convex function

It is important to understand that the functions of this family are limited to bottom with zero, but the smallest value does not have.

Example 1 - to find a maximum and minimum function at the interval \\ [\\ Mathop (Lim) _ (X \\ TO + \\ INFTY) X ^ (2N) \\) \u003d + \\ INFTY \\]

Graph (Fig. 2).

Figure 2. Schedule of the function $ F \\ left (x \\ right) \u003d x ^ (2n) $

Properties of power functions with a natural odd indicator
The definition area is all valid numbers.

$ F \\ Left (-x \\ Right) \u003d ((- x)) ^ (2n-1) \u003d (- x) ^ (2n) \u003d - f (x) $ - the function is odd.

$ F (x) $ is continuous throughout the definition area.

The area of \u200b\u200bthe value is all valid numbers.

$ f "\\ left (x \\ right) \u003d \\ left (x ^ (2N-1) \\ Right)" \u003d (2N-1) \\ Cdot X ^ (2 (N-1)) \\ GE 0 $

The function increases throughout the definition area.

$ F \\ left (x \\ right) 0 $, with $ x \\ in (0, + \\ infty) $.

$ F ("" \\ left (x \\ right)) \u003d (\\ left (\\ left (2N-1 \\ RIGHT) \\ CDOT X ^ (2 \\ left (n-1 \\ right)) \\ Right)) "\u003d 2 \\ left (2N-1 \\ RIGHT) (N-1) \\ CDOT X ^ (2N-3) $
\ \
The function is concave, with $ x \\ in (- \\ infty, 0) $ and convex, with $ x \\ in (0, + \\ infty) $.

Graph (Fig. 3).

Figure 3. Graph function $ F \\ left (x \\ right) \u003d x ^ (2N-1) $
Power function with integer
To begin with, we introduce the concept of degree with the integer.
Definition 3.

The degree of actual number $ a $ with an integer indicator $ n $ is determined by the formula:

Figure 4.
We now consider a power function with an integer, its properties and a schedule.
Definition 4.

$ F \\ left (x \\ right) \u003d x ^ n $ ($ n \\ in z) $ is called a power function with an integer.
If the degree is greater than zero, then we come to the case of a powerful function with a natural indicator. We were already considered above. For $ n \u003d 0 $, we obtain a linear function $ y \u003d 1 $. Her consideration will leave the reader. It remains to consider the properties of the power function with a negative integer
Properties of power functions with a negative integer
The definition area is $ \\ left (- \\ infty, 0 \\ right) (0, + \\ infty) $.

If the indicator is even, then the function is even, if the odd, then the function is odd.

$ F (x) $ is continuous throughout the definition area.

Value area:

If the indicator is even, then $ (0, + \\ infty) $, if odd, then $ \\ left (- \\ infty, 0 \\ right) (0, + \\ infty) $.

With an odd indicator, the function decreases, with $ x \\ in \\ left (- \\ infty, 0 \\ right) (0, + \\ infty) $. With an even indicator, the function decreases with $ x \\ in (0, + \\ infty) $. and increases, with $ x \\ in \\ left (- \\ infty, 0 \\ right) $.

$ f (x) \\ GE 0 $ over the entire field of definition

At this lesson, we will continue to study the power functions with a rational indicator, consider functions with a negative rational indicator.

1. Basic concepts and definitions

Recall the properties and graphs of power functions with a whole negative indicator.

With even N,:

Example function:

All graphs of such functions pass through two fixed points: (1; 1), (-1; 1). The feature of the functions of this species is their parity, graphics are symmetrical relative to the OU axis.

Fig. 1. Function schedule

With odd N,:

Example function:

All graphs of such functions pass through two fixed points: (1; 1), (-1; -1). The feature of the functions of this species is their oddness, graphics are symmetrical relative to the start of coordinates.

Fig. 2. Schedule function

2. Function with a negative rational indicator, graphics, properties

Recall the basic definition.

The degree of nonnegative number and a rational positive indicator is the number.

The degree of positive number and with a rational negative indicator is called the number.

Equality is performed:

For example: ; - the expression does not exist to determine the degree with a negative rational indicator; There is, since the indicator is a whole,

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

For example:

To build a graph of this feature, you can create a table. We will proceed otherwise: first we will build and study the schedule of the denominator - it is known for us (Figure 3).

Fig. 3. Function graph

The graph of the function of the denominator passes through a fixed point (1; 1). When constructing a graph of the source function, this point remains, with the root also tends to zero, the function tends to infinity. And, on the contrary, with the desire of x to infinity, the function tends to zero (Figure 4).

Fig. 4. Function schedule

Consider another feature from the family of studied functions.

It is important that by definition

Consider the schedule of the function in the denominator :, the schedule of this function is known to us, it increases on its definition area and passes through the point (1; 1) (Figure 5).

Fig. 5. Function schedule

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, on the contrary, with the desire of x to infinity, the function tends to zero (Figure 6).

Fig. 6. Function graph

Considered examples help to understand how the schedule passes and what the properties of the function being studied are functions with a negative rational indicator.

The graphs of the functions of this family pass through the point (1; 1), the function decreases throughout the definition area.

Function definition area:

The function is not limited from above, but is limited to below. The function does not have the greatest nor the smallest value.

The function is continuous, takes all the positive values \u200b\u200bfrom zero to the plus infinity.

Function convex down (Figure 15.7)

Points A and B were taken on the curve, through them a segment was taken, the whole curve is below the segment, this condition is performed for arbitrary two points on the curve, therefore, the function is convexing down. Fig. 7.

Fig. 7. Convex function

3. Solution of typical tasks

It is important to understand that the functions of this family are limited to bottom with zero, but the smallest value does not have.

Example 1 - find a maximum and minimum function at the interval)