Give a definition of a power function and give examples. Methodology for studying the topic “Properties of a power function”

Power function is a function of the form y = x p, where p is a given real number.

Properties of the power function

If the indicator p = 2n- even natural number:
- domain of definition - all real numbers, i.e. the set R;
- set of values - non-negative numbers, i.e. y ≥ 0;
- function is even;
- the function is decreasing on the interval x ≤ 0 and increasing on the interval x ≥ 0.
Example of a function with exponent p = 2n: y = x 4.

If the indicator p = 2n - 1- odd natural number:
- domain of definition - set R;
- set of values - set R;
- function is odd;
- the function is increasing on the entire real axis.
Example of a function with exponent p = 2n - 1: y = x 5.

If the indicator p = -2n, Where n- natural number:
- set of values - positive numbers y > 0;
- function is even;
- the function is increasing on the interval x 0.
Example of a function with exponent p = -2n: y = 1/x 2.

If the indicator p = -(2n - 1), Where n- natural number:
- domain of definition - set R, except x = 0;
- set of values - set R, except y = 0;
- function is odd;
- the function is decreasing on intervals x 0.
Example of a function with exponent p = -(2n - 1): y = 1/x 3.

If the indicator p- positive real non-integer number:
- domain of definition - non-negative numbers x ≥ 0;
- set of values - non-negative numbers y ≥ 0;
- the function is increasing on the interval x ≥ 0.
Example of a function with exponent p, where p is a positive real non-integer: y = x 4/3.

If the indicator p- negative real non-integer number:
- domain of definition - positive numbers x > 0;
- set of values - positive numbers y > 0;
- the function is decreasing on the interval x > 0.
Example of a function with exponent p, where p is a negative real non-integer: y = x -1/3.

Grade 10

POWER FUNCTION

Power calledfunction given by formulaWhere, p – some real number.

I . Index- an even natural number. Then the power function Wheren

D ( y )= (−; +).

2) The range of values of a function is a set of non-negative numbers, if:

set of non-positive numbers if:

3) ) . So the functionOy .

4) If, then the function decreases asX (- ; 0] and increases withX and decreases atX \[(\mathop(lim)_(x\to +\infty ) x^(2n)\ )=+\infty \]

Graph (Fig. 2).

Figure 2. Graph of the function $f\left(x\right)=x^(2n)$

Properties of a power function with a natural odd exponent

The domain of definition is all real numbers.

$f\left(-x\right)=((-x))^(2n-1)=(-x)^(2n)=-f(x)$ -- the function is odd.

$f(x)$ is continuous over the entire domain of definition.

The range is all real numbers.

$f"\left(x\right)=\left(x^(2n-1)\right)"=(2n-1)\cdot x^(2(n-1))\ge 0$

The function increases over the entire domain of definition.

$f\left(x\right)0$, for $x\in (0,+\infty)$.

$f(""\left(x\right))=(\left(\left(2n-1\right)\cdot x^(2\left(n-1\right))\right))"=2 \left(2n-1\right)(n-1)\cdot x^(2n-3)$

\ \

The function is concave for $x\in (-\infty ,0)$ and convex for $x\in (0,+\infty)$.

Graph (Fig. 3).

Figure 3. Graph of the function $f\left(x\right)=x^(2n-1)$

Power function with integer exponent

First, let's introduce the concept of a degree with an integer exponent.

Definition 3

The power of a real number $a$ with integer exponent $n$ is determined by the formula:

Figure 4.

Let us now consider a power function with an integer exponent, its properties and graph.

Definition 4

$f\left(x\right)=x^n$ ($n\in Z)$ is called a power function with an integer exponent.

If the degree is greater than zero, then we come to the case of a power function with a natural exponent. We have already discussed it above. For $n=0$ we get a linear function $y=1$. We will leave its consideration to the reader. It remains to consider the properties of a power function with a negative integer exponent

Properties of a power function with a negative integer exponent

The domain of definition is $\left(-\infty ,0\right)(0,+\infty)$.

If the exponent is even, then the function is even; if it is odd, then the function is odd.

$f(x)$ is continuous over the entire domain of definition.

Scope:

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

For an odd exponent, the function decreases as $x\in \left(-\infty ,0\right)(0,+\infty)$. For an even exponent, the function decreases as $x\in (0,+\infty)$. and increases as $x\in \left(-\infty ,0\right)$.

$f(x)\ge 0$ over the entire domain of definition

On the domain of definition of the power function y = x p the following formulas hold:
; ;
;
; ;
; ;
; .

Properties of power functions and their graphs

Power function with exponent equal to zero, p = 0

If the exponent of the power function y = x p is equal to zero, p = 0, then the power function is defined for all x ≠ 0 and is a constant equal to one:
y = x p = x 0 = 1, x ≠ 0.

Power function with natural odd exponent, p = n = 1, 3, 5, ...

Consider a power function y = x p = x n with a natural odd exponent n = 1, 3, 5, ... .

This indicator can also be written in the form: n = 2k + 1, where k = 0, 1, 2, 3, ... is a non-negative integer. Below are the properties and graphs of such functions. Graph of a power function y = x n with a natural odd exponent at different meanings

exponent n = 1, 3, 5, ... . -∞ < x < ∞
Domain: -∞ < y < ∞
Multiple meanings: Parity:
odd, y(-x) = - y(x) Monotone:
monotonically increases Extremes:
No
Convex:< x < 0 выпукла вверх
at -∞< x < ∞ выпукла вниз
at 0 Inflection points:
Inflection points:
x = 0, y = 0
;
Limits:
Private values:
at x = -1,
y(-1) = (-1) n ≡ (-1) 2k+1 = -1
at x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = 1, the function is its inverse: x = y

for n ≠ 1, the inverse function is the root of degree n:

Power function with natural even exponent, p = n = 2, 4, 6, ...

Consider a power function y = x p = x n with a natural even exponent n = 2, 4, 6, ... .

exponent n = 1, 3, 5, ... . -∞ < x < ∞
Domain: This indicator can also be written in the form: n = 2k, where k = 1, 2, 3, ... - natural. The properties and graphs of such functions are given below.< ∞
Multiple meanings: Graph of a power function y = x n with a natural even exponent for various values of the exponent n = 2, 4, 6, ....
odd, y(-x) = - y(x)
0 ≤ y
even, y(-x) = y(x)
monotonically increases for x ≤ 0 monotonically decreases
No for x ≥ 0 monotonically increases
at 0 Extremes:
minimum, x = 0, y = 0 Inflection points:
x = 0, y = 0
;
Limits:
convex down Intersection points with coordinate axes:
y(-1) = (-1) n ≡ (-1) 2k+1 = -1
at x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
at x = -1, y(-1) = (-1) n ≡ (-1) 2k = 1:
for n = 2,

Square root

for n ≠ 2, root of degree n:

Power function with negative integer exponent, p = n = -1, -2, -3, ...

Consider a power function y = x p = x n with an integer negative exponent n = -1, -2, -3, ... .

If we put n = -k, where k = 1, 2, 3, ... is a natural number, then it can be represented as:

exponent n = 1, 3, 5, ... . Graph of a power function y = x n with a negative integer exponent for various values of the exponent n = -1, -2, -3, ....
Domain: Odd exponent, n = -1, -3, -5, ...
Multiple meanings: Parity:
odd, y(-x) = - y(x) Below are the properties of the function y = x n with an odd negative exponent n = -1, -3, -5, ....
monotonically increases Extremes:
No
x ≠ 0< 0 : выпукла вверх
y ≠ 0
at 0 Extremes:
minimum, x = 0, y = 0 Extremes:
monotonically decreases
x ≠ 0< 0, y < 0
at x
x = 0, y = 0
; ; ;
Limits:
at x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
for x > 0: convex downward
Sign:< -2 ,

for x > 0, y > 0

when n = -1,

exponent n = 1, 3, 5, ... . Graph of a power function y = x n with a negative integer exponent for various values of the exponent n = -1, -2, -3, ....
Domain: at n
Multiple meanings: Graph of a power function y = x n with a natural even exponent for various values of the exponent n = 2, 4, 6, ....
odd, y(-x) = - y(x)
x ≠ 0< 0 : монотонно возрастает
Even exponent, n = -2, -4, -6, ...
monotonically increases Extremes:
No for x ≥ 0 monotonically increases
at 0 Extremes:
minimum, x = 0, y = 0 Extremes:
monotonically decreases at n
x = 0, y = 0
; ; ;
Limits:
at x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
Below are the properties of the function y = x n with an even negative exponent n = -2, -4, -6, ....
Sign:< -2 ,

y > 0

Consider a power function y = x p with a rational (fractional) exponent, where n is an integer, m > 1 is a natural number. Moreover, n, m do not have common divisors.

The denominator of the fractional indicator is odd

Let the denominator of the fractional exponent be odd: m = 3, 5, 7, ... . In this case, the power function x p is defined for both positive and negative values of the argument x.

Let us consider the properties of such power functions when the exponent p is within certain limits.< 0

The p-value is negative, p Let the rational exponent (with odd denominator m = 3, 5, 7, ...): .

less than zero

Graphs of power functions with a rational negative exponent for various values of the exponent, where m = 3, 5, 7, ... - odd.

Odd numerator, n = -1, -3, -5, ...

exponent n = 1, 3, 5, ... . Graph of a power function y = x n with a negative integer exponent for various values of the exponent n = -1, -2, -3, ....
Domain: Odd exponent, n = -1, -3, -5, ...
Multiple meanings: Parity:
odd, y(-x) = - y(x) Below are the properties of the function y = x n with an odd negative exponent n = -1, -3, -5, ....
monotonically increases Extremes:
No
x ≠ 0< 0 : выпукла вверх
y ≠ 0
at 0 Extremes:
minimum, x = 0, y = 0 Extremes:
monotonically decreases
x ≠ 0< 0, y < 0
at x
x = 0, y = 0
; ; ;
Limits:
We present the properties of the power function y = x p with a rational negative exponent, where n = -1, -3, -5, ... is an odd negative integer, m = 3, 5, 7 ... is an odd natural integer.
at x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1

at x = -1, y(-1) = (-1) n = -1

Even numerator, n = -2, -4, -6, ...

exponent n = 1, 3, 5, ... . Graph of a power function y = x n with a negative integer exponent for various values of the exponent n = -1, -2, -3, ....
Domain: at n
Multiple meanings: Graph of a power function y = x n with a natural even exponent for various values of the exponent n = 2, 4, 6, ....
odd, y(-x) = - y(x)
x ≠ 0< 0 : монотонно возрастает
Even exponent, n = -2, -4, -6, ...
monotonically increases Extremes:
No for x ≥ 0 monotonically increases
at 0 Extremes:
minimum, x = 0, y = 0 Extremes:
monotonically decreases at n
x = 0, y = 0
; ; ;
Limits:
Properties of the power function y = x p with a rational negative exponent, where n = -2, -4, -6, ... is an even negative integer, m = 3, 5, 7 ... is an odd natural integer.
at x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1

at x = -1, y(-1) = (-1) n = 1< p < 1

The p-value is positive, less than one, 0< p < 1 ) при различных значениях показателя степени , где m = 3, 5, 7, ... - нечетное.

Graph of a power function with rational exponent (0

< p < 1 , где n = 1, 3, 5, ... - нечетное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

exponent n = 1, 3, 5, ... . -∞ < x < +∞
Domain: -∞ < y < +∞
Multiple meanings: Parity:
odd, y(-x) = - y(x) Monotone:
monotonically increases Extremes:
No
x ≠ 0< 0 : выпукла вниз
Odd numerator, n = 1, 3, 5, ...
at 0 Inflection points:
minimum, x = 0, y = 0 Inflection points:
monotonically decreases
x ≠ 0< 0, y < 0
at x
x = 0, y = 0
;
Limits:
for x > 0: convex upward
at x = -1, y(-1) = -1
at x = 0, y(0) = 0
for x = 1, y(1) = 1 n = 1

for x = 1, y(1) = 1

Even numerator, n = 2, 4, 6, ...< p < 1 , где n = 2, 4, 6, ... - четное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

exponent n = 1, 3, 5, ... . -∞ < x < +∞
Domain: This indicator can also be written in the form: n = 2k, where k = 1, 2, 3, ... - natural. The properties and graphs of such functions are given below.< +∞
Multiple meanings: Graph of a power function y = x n with a natural even exponent for various values of the exponent n = 2, 4, 6, ....
odd, y(-x) = - y(x)
x ≠ 0< 0 : монотонно убывает
The properties of the power function y = x p with a rational exponent within 0 are presented
monotonically increases for x > 0: increases monotonically
No minimum at x = 0, y = 0
at 0 Extremes:
minimum, x = 0, y = 0 Inflection points:
monotonically decreases convex upward for x ≠ 0
x = 0, y = 0
;
Limits:
for x ≠ 0, y > 0
at x = -1, y(-1) = -1
at x = 0, y(0) = 0
for x = 1, y(1) = 1 n = 1

at x = -1, y(-1) = 1

The index p is greater than one, p > 1

Graph of a power function with a rational exponent (p > 1) for various values of the exponent, where m = 3, 5, 7, ... - odd.

Odd numerator, n = 5, 7, 9, ...

exponent n = 1, 3, 5, ... . -∞ < x < ∞
Domain: -∞ < y < ∞
Multiple meanings: Parity:
odd, y(-x) = - y(x) Monotone:
monotonically increases Extremes:
No
Convex:< x < 0 выпукла вверх
at -∞< x < ∞ выпукла вниз
at 0 Inflection points:
minimum, x = 0, y = 0 Inflection points:
x = 0, y = 0
;
Limits:
for x > 0: convex upward
at x = -1, y(-1) = -1
at x = 0, y(0) = 0
for x = 1, y(1) = 1 n = 1

Properties of the power function y = x p with a rational exponent greater than one: .

Where n = 5, 7, 9, ... - odd natural, m = 3, 5, 7 ... - odd natural.

exponent n = 1, 3, 5, ... . -∞ < x < ∞
Domain: This indicator can also be written in the form: n = 2k, where k = 1, 2, 3, ... - natural. The properties and graphs of such functions are given below.< ∞
Multiple meanings: Graph of a power function y = x n with a natural even exponent for various values of the exponent n = 2, 4, 6, ....
odd, y(-x) = - y(x)
x ≠ 0< 0 монотонно убывает
Even numerator, n = 4, 6, 8, ...
monotonically increases for x > 0: increases monotonically
No for x ≥ 0 monotonically increases
at 0 Extremes:
minimum, x = 0, y = 0 Inflection points:
x = 0, y = 0
;
Limits:
for x ≠ 0, y > 0
at x = -1, y(-1) = -1
at x = 0, y(0) = 0
for x = 1, y(1) = 1 n = 1

Properties of the power function y = x p with a rational exponent greater than one: .

Let the denominator of the fractional exponent be even: m = 2, 4, 6, ... . In this case, the power function x p is not defined for negative values of the argument. Its properties coincide with the properties of a power function with an irrational exponent (see the next section).

Power function with irrational exponent

Consider a power function y = x p with an irrational exponent p.

The properties of such functions differ from those discussed above in that they are not defined for negative values of the argument x.

For positive values of the argument, the properties depend only on the value of the exponent p and do not depend on whether p is integer, rational, or irrational.< 0

exponent n = 1, 3, 5, ... . y = x p for different values of the exponent p.
Domain: at n
odd, y(-x) = - y(x) Below are the properties of the function y = x n with an odd negative exponent n = -1, -3, -5, ....
No for x ≥ 0 monotonically increases
at 0 Extremes:
minimum, x = 0, y = 0 Extremes:
x = 0, y = 0 ;
Power function with negative exponent p x > 0

Private meaning:

For x = 1, y(1) = 1 p = 1< p < 1

exponent n = 1, 3, 5, ... . Power function with positive exponent p > 0
Domain: Indicator less than one 0
odd, y(-x) = - y(x) Monotone:
No x ≥ 0
at 0 Extremes:
minimum, x = 0, y = 0 Inflection points:
x = 0, y = 0
Limits: y ≥ 0
x > 0

convex upward

exponent n = 1, 3, 5, ... . Power function with positive exponent p > 0
Domain: Indicator less than one 0
odd, y(-x) = - y(x) Monotone:
No for x ≥ 0 monotonically increases
at 0 Extremes:
minimum, x = 0, y = 0 Inflection points:
x = 0, y = 0
Limits: y ≥ 0
x > 0

For x = 0, y(0) = 0 p = 0 .
The indicator is greater than one p > 1

References:

I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

Power function

Definition 5

The power function is defined by the formula y = x a.

The appearance of the graphs and the properties of the function depend on the value of the exponent.

when a power function has whole indicator a, then the type of graph of the power function and its properties depend on whether the exponent is even or odd, as well as what sign the exponent has. Let's consider all these special cases in more detail below;
the exponent can be fractional or irrational - depending on this, the type of graphs and properties of the function also vary. We will analyze special cases by setting several conditions: 0< a < 1 ; a > 1 ; - 1 < a < 0 и a < - 1 ;
a power function can have a zero exponent; we will also analyze this case in more detail below.

Let's analyze the power function y = x a, when a is an odd positive number, for example, a = 1, 3, 5...

For clarity, we indicate the graphs of such power functions: y = x (graphic color black), y = x 3 (blue color of the graph), y = x 5 (red color of the graph), y = x 7 (graphic color green). When a = 1, we get the linear function y = x.

Definition 6

Properties of a power function when the exponent is odd positive

the function is increasing for x ∈ (- ∞ ; + ∞) ;
the function has convexity for x ∈ (- ∞ ; 0 ] and concavity for x ∈ [ 0 ; + ∞) (excluding the linear function);
the inflection point has coordinates (0 ; 0) (excluding linear function);
there are no asymptotes;
points of passage of the function: (- 1 ; - 1) , (0 ; 0) , (1 ; 1) .

Let's analyze the power function y = x a, when a is an even positive number, for example, a = 2, 4, 6...

For clarity, we indicate the graphs of such power functions: y = x 2 (graphic color black), y = x 4 (blue color of the graph), y = x 8 (red color of the graph). When a = 2, we obtain a quadratic function, the graph of which is a quadratic parabola.

Definition 7

Properties of a power function when the exponent is even positive:

domain of definition: x ∈ (- ∞ ; + ∞) ;
decreasing for x ∈ (- ∞ ; 0 ] ;
the function has concavity for x ∈ (- ∞ ; + ∞) ;
there are no inflection points;
there are no asymptotes;
points of passage of the function: (- 1 ; 1) , (0 ; 0) , (1 ; 1) .

The figure below shows examples of power function graphs y = x a when a is an odd negative number: y = x - 9 (graphic color black); y = x - 5 (blue color of the graph); y = x - 3 (red color of the graph); y = x - 1 (graphic color green). When a = - 1, we obtain inverse proportionality, the graph of which is a hyperbola.

Definition 8

Properties of a power function when the exponent is odd negative:

When x = 0, we obtain a discontinuity of the second kind, since lim x → 0 - 0 x a = - ∞, lim x → 0 + 0 x a = + ∞ for a = - 1, - 3, - 5, …. Thus, the straight line x = 0 is a vertical asymptote;

range: y ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;
the function is odd because y (- x) = - y (x);
the function is decreasing for x ∈ - ∞ ; 0 ∪ (0 ; + ∞) ;
the function has convexity for x ∈ (- ∞ ; 0) and concavity for x ∈ (0 ; + ∞) ;
there are no inflection points;

k = lim x → ∞ x a x = 0, b = lim x → ∞ (x a - k x) = 0 ⇒ y = k x + b = 0, when a = - 1, - 3, - 5, . . . .

points of passage of the function: (- 1 ; - 1) , (1 ; 1) .

The figure below shows examples of graphs of the power function y = x a when a is an even negative number: y = x - 8 (graphic color black); y = x - 4 (blue color of the graph); y = x - 2 (red color of the graph).

Definition 9

Properties of a power function when the exponent is even negative:

domain of definition: x ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;

When x = 0, we obtain a discontinuity of the second kind, since lim x → 0 - 0 x a = + ∞, lim x → 0 + 0 x a = + ∞ for a = - 2, - 4, - 6, …. Thus, the straight line x = 0 is a vertical asymptote;

the function is even because y(-x) = y(x);
the function is increasing for x ∈ (- ∞ ; 0) and decreasing for x ∈ 0; + ∞ ;
the function has concavity at x ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;
there are no inflection points;
horizontal asymptote – straight line y = 0, because:

k = lim x → ∞ x a x = 0 , b = lim x → ∞ (x a - k x) = 0 ⇒ y = k x + b = 0 when a = - 2 , - 4 , - 6 , . . . .

points of passage of the function: (- 1 ; 1) , (1 ; 1) .

From the very beginning, pay attention to the following aspect: in the case where a – positive fraction with an odd denominator, some authors take the interval - ∞ as the domain of definition of this power function; + ∞ , stipulating that the exponent a is an irreducible fraction. On this moment The authors of many educational publications on algebra and principles of analysis DO NOT DEFINE power functions, where the exponent is a fraction with an odd denominator for negative values of the argument. Further we will adhere to exactly this position: we will take the set [ 0 ; + ∞) . Recommendation for students: find out the teacher’s view on this point in order to avoid disagreements.

So, let's look at the power function y = x a , when the exponent is rational or irrational number provided that 0< a < 1 .

Let us illustrate the power functions with graphs y = x a when a = 11 12 (graphic color black); a = 5 7 (red color of the graph); a = 1 3 (blue color of the graph); a = 2 5 (green color of the graph).

Other values of the exponent a (provided 0< a < 1) дадут аналогичный вид графика.

Definition 10

Properties of the power function at 0< a < 1:

range: y ∈ [ 0 ; + ∞) ;
the function is increasing for x ∈ [ 0 ; + ∞) ;
the function is convex for x ∈ (0 ; + ∞);
there are no inflection points;
there are no asymptotes;

Let's analyze the power function y = x a, when the exponent is a non-integer rational or irrational number, provided that a > 1.

Let us illustrate with graphs the power function y = x a under given conditions using the following functions as an example: y = x 5 4 , y = x 4 3 , y = x 7 3 , y = x 3 π (black, red, blue, green graphs, respectively).

Other values of the exponent a, provided a > 1, will give a similar graph.

Definition 11

Properties of the power function for a > 1:

domain of definition: x ∈ [ 0 ; + ∞) ;
range: y ∈ [ 0 ; + ∞) ;
this function is a function of general form (it is neither odd nor even);
the function is increasing for x ∈ [ 0 ; + ∞) ;
the function has concavity for x ∈ (0 ; + ∞) (when 1< a < 2) и выпуклость при x ∈ [ 0 ; + ∞) (когда a > 2);
there are no inflection points;
there are no asymptotes;
passing points of the function: (0 ; 0) , (1 ; 1) .

Please note! When a is a negative fraction with an odd denominator, in the works of some authors there is a view that the domain of definition is in in this case– interval - ∞; 0 ∪ (0 ; + ∞) with the caveat that the exponent a is an irreducible fraction. Currently the authors educational materials in algebra and principles of analysis DO NOT DETERMINE power functions with an exponent in the form of a fraction with an odd denominator for negative values of the argument. Further, we adhere to exactly this view: we take the set (0 ; + ∞) as the domain of definition of power functions with fractional negative exponents. Recommendation for students: Clarify your teacher's vision at this point to avoid disagreements.

Let's continue the topic and analyze the power function y = x a provided: - 1< a < 0 .

Let us present a drawing of graphs of the following functions: y = x - 5 6, y = x - 2 3, y = x - 1 2 2, y = x - 1 7 (black, red, blue, green color of the lines, respectively).

Definition 12

Properties of the power function at - 1< a < 0:

lim x → 0 + 0 x a = + ∞ when - 1< a < 0 , т.е. х = 0 – вертикальная асимптота;

range: y ∈ 0 ; + ∞ ;
this function is a function of general form (it is neither odd nor even);
there are no inflection points;

The drawing below shows graphs of power functions y = x - 5 4, y = x - 5 3, y = x - 6, y = x - 24 7 (black, red, blue, green colors curves respectively).

Definition 13

Properties of the power function for a< - 1:

domain of definition: x ∈ 0 ; + ∞ ;

lim x → 0 + 0 x a = + ∞ when a< - 1 , т.е. х = 0 – вертикальная асимптота;

range: y ∈ (0 ; + ∞) ;
this function is a function of general form (it is neither odd nor even);
the function is decreasing for x ∈ 0; + ∞ ;
the function has a concavity for x ∈ 0; + ∞ ;
there are no inflection points;
horizontal asymptote – straight line y = 0;
point of passage of the function: (1; 1) .

When a = 0 and x ≠ 0, we obtain the function y = x 0 = 1, which defines the line from which the point (0; 1) is excluded (it was agreed that the expression 0 0 will not be given any meaning).

The exponential function has the form y = a x, where a > 0 and a ≠ 1, and the graph of this function looks different based on the value of the base a. Let's consider special cases.

First, let's look at the situation when the base exponential function has a value from zero to one (0< a < 1) . A clear example graphs of functions for a = 1 2 (blue color of the curve) and a = 5 6 (red color of the curve) will serve.

The graphs of the exponential function will have a similar appearance for other values of the base under the condition 0< a < 1 .

Definition 14

Properties of the exponential function when the base is less than one:

range: y ∈ (0 ; + ∞) ;
this function is a function of general form (it is neither odd nor even);
an exponential function whose base is less than one is decreasing over the entire domain of definition;
there are no inflection points;
horizontal asymptote – straight line y = 0 with variable x tending to + ∞;

Now consider the case when the base of the exponential function is greater than one (a > 1).

Let us illustrate this special case with a graph of exponential functions y = 3 2 x (blue color of the curve) and y = e x (red color of the graph).

Other values of the base, larger units, will give a similar appearance to the graph of the exponential function.

Definition 15

Properties of the exponential function when the base is greater than one:

domain of definition – the entire set of real numbers;
range: y ∈ (0 ; + ∞) ;
this function is a function of general form (it is neither odd nor even);
an exponential function whose base is greater than one is increasing as x ∈ - ∞; + ∞ ;
the function has a concavity at x ∈ - ∞; + ∞ ;
there are no inflection points;
horizontal asymptote – straight line y = 0 with variable x tending to - ∞;
point of passage of the function: (0; 1) .

The logarithmic function has the form y = log a (x), where a > 0, a ≠ 1.

This function is only defined when positive values argument: for x ∈ 0 ; + ∞ .

The graph of a logarithmic function has different kind, based on the value of base a.

Let us first consider the situation when 0< a < 1 . Продемонстрируем этот частный случай графиком логарифмической функции при a = 1 2 (синий цвет кривой) и а = 5 6 (красный цвет кривой).

Other values of the base, not larger units, will give a similar type of graph.

Definition 16

Properties of a logarithmic function when the base is less than one:

domain of definition: x ∈ 0 ; + ∞ . As x tends to zero from the right, the function values tend to +∞;
range: y ∈ - ∞ ; + ∞ ;
this function is a function of general form (it is neither odd nor even);
logarithmic
the function has a concavity for x ∈ 0; + ∞ ;
there are no inflection points;
there are no asymptotes;

Now let's look at the special case when the base of the logarithmic function is greater than one: a > 1 . The drawing below shows graphs of logarithmic functions y = log 3 2 x and y = ln x (blue and red colors of the graphs, respectively).

Other values of the base greater than one will give a similar type of graph.

Definition 17

Properties of a logarithmic function when the base is greater than one:

domain of definition: x ∈ 0 ; + ∞ . As x tends to zero from the right, the function values tend to - ∞ ;
range: y ∈ - ∞ ; + ∞ (the entire set of real numbers);
this function is a function of general form (it is neither odd nor even);
the logarithmic function is increasing for x ∈ 0; + ∞ ;
the function is convex for x ∈ 0; + ∞ ;
there are no inflection points;
there are no asymptotes;
point of passage of the function: (1; 0) .

The trigonometric functions are sine, cosine, tangent and cotangent. Let's look at the properties of each of them and the corresponding graphics.

In general, all trigonometric functions are characterized by the property of periodicity, i.e. when function values are repeated at different meanings arguments differing from each other by the period f (x + T) = f (x) (T – period). Thus, the item “smallest positive period” is added to the list of properties of trigonometric functions. In addition, we will indicate the values of the argument at which the corresponding function becomes zero.

Sine function: y = sin(x)

The graph of this function is called a sine wave.

Definition 18

Properties of the sine function:

domain of definition: the entire set of real numbers x ∈ - ∞ ; + ∞ ;
the function vanishes when x = π · k, where k ∈ Z (Z is the set of integers);
the function is increasing for x ∈ - π 2 + 2 π · k ; π 2 + 2 π · k, k ∈ Z and decreasing for x ∈ π 2 + 2 π · k; 3 π 2 + 2 π · k, k ∈ Z;
the sine function has local maxima at points π 2 + 2 π · k; 1 and local minima at points - π 2 + 2 π · k; - 1, k ∈ Z;
the sine function is concave when x ∈ - π + 2 π · k ; 2 π · k, k ∈ Z and convex when x ∈ 2 π · k; π + 2 π k, k ∈ Z;
there are no asymptotes.

Cosine function: y = cos(x)

The graph of this function is called a cosine wave.

Definition 19

Properties of the cosine function:

domain of definition: x ∈ - ∞ ; + ∞ ;
smallest positive period: T = 2 π;
range of values: y ∈ - 1 ; 1 ;
this function is even, since y (- x) = y (x);
the function is increasing for x ∈ - π + 2 π · k ; 2 π · k, k ∈ Z and decreasing for x ∈ 2 π · k; π + 2 π k, k ∈ Z;
the cosine function has local maxima at points 2 π · k ; 1, k ∈ Z and local minima at points π + 2 π · k; - 1, k ∈ z;
the cosine function is concave when x ∈ π 2 + 2 π · k ; 3 π 2 + 2 π · k , k ∈ Z and convex when x ∈ - π 2 + 2 π · k ; π 2 + 2 π · k, k ∈ Z;
inflection points have coordinates π 2 + π · k; 0 , k ∈ Z
there are no asymptotes.

Tangent function: y = t g (x)

The graph of this function is called tangent.

Definition 20

Properties of the tangent function:

domain of definition: x ∈ - π 2 + π · k ; π 2 + π · k, where k ∈ Z (Z is the set of integers);
Behavior of the tangent function on the boundary of the domain of definition lim x → π 2 + π · k + 0 t g (x) = - ∞ , lim x → π 2 + π · k - 0 t g (x) = + ∞ . Thus, the straight lines x = π 2 + π · k k ∈ Z are vertical asymptotes;
the function vanishes when x = π · k for k ∈ Z (Z is the set of integers);
range: y ∈ - ∞ ; + ∞ ;
this function is odd, since y (- x) = - y (x) ;
the function is increasing as - π 2 + π · k ; π 2 + π · k, k ∈ Z;
the tangent function is concave for x ∈ [π · k; π 2 + π · k) , k ∈ Z and convex for x ∈ (- π 2 + π · k ; π · k ] , k ∈ Z ;
inflection points have coordinates π · k ; 0 , k ∈ Z ;

Cotangent function: y = c t g (x)

The graph of this function is called a cotangentoid. .

Definition 21

Properties of the cotangent function:

domain of definition: x ∈ (π · k ; π + π · k) , where k ∈ Z (Z is the set of integers);

Behavior of the cotangent function on the boundary of the domain of definition lim x → π · k + 0 t g (x) = + ∞ , lim x → π · k - 0 t g (x) = - ∞ . Thus, the straight lines x = π · k k ∈ Z are vertical asymptotes;

smallest positive period: T = π;
the function vanishes when x = π 2 + π · k for k ∈ Z (Z is the set of integers);
range: y ∈ - ∞ ; + ∞ ;
this function is odd, since y (- x) = - y (x) ;
the function is decreasing for x ∈ π · k ; π + π k, k ∈ Z;
the cotangent function is concave for x ∈ (π · k; π 2 + π · k ], k ∈ Z and convex for x ∈ [ - π 2 + π · k ; π · k), k ∈ Z ;
inflection points have coordinates π 2 + π · k; 0 , k ∈ Z ;
There are no oblique or horizontal asymptotes.

The inverse trigonometric functions are arcsine, arccosine, arctangent and arccotangent. Often, due to the presence of the prefix “arc” in the name, inverse trigonometric functions are called arc functions .

Arc sine function: y = a r c sin (x)

Definition 22

Properties of the arcsine function:

this function is odd, since y (- x) = - y (x) ;
the arcsine function has a concavity for x ∈ 0; 1 and convexity for x ∈ - 1 ; 0 ;
inflection points have coordinates (0; 0), which is also the zero of the function;
there are no asymptotes.

Arc cosine function: y = a r c cos (x)

Definition 23

Properties of the arc cosine function:

domain of definition: x ∈ - 1 ; 1 ;
range: y ∈ 0 ; π;
this function is of a general form (neither even nor odd);
the function is decreasing over the entire domain of definition;
the arc cosine function has a concavity at x ∈ - 1; 0 and convexity for x ∈ 0; 1 ;
inflection points have coordinates 0; π 2;
there are no asymptotes.

Arctangent function: y = a r c t g (x)

Definition 24

Properties of the arctangent function:

domain of definition: x ∈ - ∞ ; + ∞ ;
range of values: y ∈ - π 2 ; π 2;
this function is odd, since y (- x) = - y (x) ;
the function is increasing over the entire domain of definition;
the arctangent function has concavity for x ∈ (- ∞ ; 0 ] and convexity for x ∈ [ 0 ; + ∞);
the inflection point has coordinates (0; 0), which is also the zero of the function;
horizontal asymptotes are straight lines y = - π 2 as x → - ∞ and y = π 2 as x → + ∞ (in the figure, the asymptotes are green lines).

Arc tangent function: y = a r c c t g (x)

Definition 25

Properties of the arccotangent function:

domain of definition: x ∈ - ∞ ; + ∞ ;
range: y ∈ (0; π) ;
this function is of a general form;
the function is decreasing over the entire domain of definition;
the arc cotangent function has a concavity for x ∈ [ 0 ; + ∞) and convexity for x ∈ (- ∞ ; 0 ] ;
the inflection point has coordinates 0; π 2;
horizontal asymptotes are straight lines y = π at x → - ∞ (green line in the drawing) and y = 0 at x → + ∞.

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Give a definition of a power function and give examples. Methodology for studying the topic “Properties of a power function”

Properties of the power function

Properties of a power function with a natural odd exponent

Power function with integer exponent

Properties of a power function with a negative integer exponent

Properties of power functions and their graphs

Power function with exponent equal to zero, p = 0

Power function with natural odd exponent, p = n = 1, 3, 5, ...

for n ≠ 1, the inverse function is the root of degree n:

Square root

Consider a power function y = x p = x n with an integer negative exponent n = -1, -2, -3, ... .

for x > 0, y > 0

y > 0

The denominator of the fractional indicator is odd

Let us consider the properties of such power functions when the exponent p is within certain limits.< 0

Graphs of power functions with a rational negative exponent for various values ​​of the exponent, where m = 3, 5, 7, ... - odd.

at x = -1, y(-1) = (-1) n = -1

at x = -1, y(-1) = (-1) n = 1< p < 1

Graph of a power function with rational exponent (0

for x = 1, y(1) = 1

at x = -1, y(-1) = 1

Graph of a power function with a rational exponent (p > 1) for various values ​​of the exponent, where m = 3, 5, 7, ... - odd.

Properties of the power function y = x p with a rational exponent greater than one: .

Properties of the power function y = x p with a rational exponent greater than one: .

Power function with irrational exponent

For positive values ​​of the argument, the properties depend only on the value of the exponent p and do not depend on whether p is integer, rational, or irrational.< 0

Private meaning:

For x = 1, y(1) = 1 p = 1< p < 1

convex upward

Power function

Graphs of power functions with a rational negative exponent for various values of the exponent, where m = 3, 5, 7, ... - odd.

Graph of a power function with a rational exponent (p > 1) for various values of the exponent, where m = 3, 5, 7, ... - odd.

For positive values of the argument, the properties depend only on the value of the exponent p and do not depend on whether p is integer, rational, or irrational.< 0