1 power function graph property. An exponential function - properties, graphs, formulas. Properties of the cosine function

Are you familiar with the features y=x, y=x 2 , y=x 3 , y=1/x etc. All these functions are special cases of the power function, i.e., the function y=xp, where p is a given real number.
The properties and graph of a power function essentially depend on the properties of a power with a real exponent, and in particular on the values ​​for which x And p makes sense x p. Let us proceed to a similar consideration of various cases, depending on
exponent p.

  1. Indicator p=2n is an even natural number.
y=x2n, where n is a natural number, has the following

properties:

  • the domain of definition is all real numbers, i.e., the set R;
  • set of values ​​- non-negative numbers, i.e. y is greater than or equal to 0;
  • function y=x2n even, because x 2n=(- x) 2n
  • the function is decreasing on the interval x<0 and increasing on the interval x>0.
Function Graph y=x2n has the same form as, for example, the graph of a function y=x4.

2. Indicator p=2n-1- odd natural number
In this case, the power function y=x 2n-1, where is a natural number, has the following properties:

  • domain of definition - set R;
  • set of values ​​- set R;
  • function y=x 2n-1 odd because (- x) 2n-1=x 2n-1 ;
  • the function is increasing on the entire real axis.
Function Graph y=x 2n-1 has the same form as, for example, the graph of the function y=x 3 .

3.Indicator p=-2n, where n- natural number.

In this case, the power function y=x -2n=1/x2n has the following properties:

  • domain of definition - set R, except for x=0;
  • set of values ​​- positive numbers y>0;
  • function y =1/x2n even, because 1/(-x) 2n=1/x2n;
  • the function is increasing on the interval x<0 и убывающей на промежутке x>0.
Graph of the function y =1/x2n has the same form as, for example, the graph of the function y =1/x2.

On the domain 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 constant, equal to one:
y \u003d x p \u003d x 0 \u003d 1, x ≠ 0.

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

Consider a power function y = x p = x n with natural odd exponent n = 1, 3, 5, ... . Such an indicator can also be written as: 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 for various values ​​of the exponent n = 1, 3, 5, ... .

Domain: -∞ < x < ∞
Multiple values: -∞ < y < ∞
Parity: odd, y(-x) = - y(x)
Monotone: increases monotonically
Extremes: No
Convex:
at -∞< x < 0 выпукла вверх
at 0< x < ∞ выпукла вниз
Breakpoints: x=0, y=0
x=0, y=0
Limits:
;
Private values:
at x = -1,
y(-1) = (-1) n ≡ (-1) 2k+1 = -1
for x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = 1 , the function is inverse to itself: x = y
for n ≠ 1, the inverse function is a 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 natural even exponent n = 2, 4, 6, ... . Such an indicator can also be written as: n = 2k, where k = 1, 2, 3, ... is a natural number. The properties and graphs of such functions are given below.

Graph of a power function y = x n with a natural even exponent for various values ​​of the exponent n = 2, 4, 6, ... .

Domain: -∞ < x < ∞
Multiple values: 0 ≤ y< ∞
Parity: even, y(-x) = y(x)
Monotone:
for x ≤ 0 monotonically decreases
for x ≥ 0 monotonically increases
Extremes: minimum, x=0, y=0
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Limits:
;
Private values:
for x = -1, y(-1) = (-1) n ≡ (-1) 2k = 1
for x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = 2, square root:
for n ≠ 2, root of degree n:

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

Consider a power function y = x p = x n with a negative integer 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:

Graph of a power function y = x n with a negative integer exponent for various values ​​of the exponent n = -1, -2, -3, ... .

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

Below are the properties of the function y = x n with an odd negative exponent n = -1, -3, -5, ... .

Domain: x ≠ 0
Multiple values: y ≠ 0
Parity: odd, y(-x) = - y(x)
Monotone: decreases monotonically
Extremes: No
Convex:
at x< 0 : выпукла вверх
for x > 0 : convex down
Breakpoints: No
Intersection points with coordinate axes: No
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
; ; ;
Private values:
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = -1,
for n< -2 ,

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

Below are the properties of the function y = x n with an even negative exponent n = -2, -4, -6, ... .

Domain: x ≠ 0
Multiple values: y > 0
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 : монотонно возрастает
for x > 0 : monotonically decreasing
Extremes: No
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: No
Sign: y > 0
Limits:
; ; ;
Private values:
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = -2,
for n< -2 ,

Power function with rational (fractional) exponent

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 x values. Consider the properties of such power functions when the exponent p is within certain limits.

p is negative, p< 0

Let the rational exponent (with odd denominator m = 3, 5, 7, ... ) be less than zero: .

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

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

Here are 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 number.

Domain: x ≠ 0
Multiple values: y ≠ 0
Parity: odd, y(-x) = - y(x)
Monotone: decreases monotonically
Extremes: No
Convex:
at x< 0 : выпукла вверх
for x > 0 : convex down
Breakpoints: No
Intersection points with coordinate axes: No
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
; ; ;
Private values:
for x = -1, y(-1) = (-1) n = -1
for x = 1, y(1) = 1 n = 1
Reverse function:

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

Properties of a 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 number.

Domain: x ≠ 0
Multiple values: y > 0
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 : монотонно возрастает
for x > 0 : monotonically decreasing
Extremes: No
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: No
Sign: y > 0
Limits:
; ; ;
Private values:
for x = -1, y(-1) = (-1) n = 1
for x = 1, y(1) = 1 n = 1
Reverse function:

The p-value is positive, less than one, 0< p < 1

Graph of a power function with a rational exponent (0< p < 1 ) при различных значениях показателя степени , где m = 3, 5, 7, ... - нечетное.

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

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

Domain: -∞ < x < +∞
Multiple values: -∞ < y < +∞
Parity: odd, y(-x) = - y(x)
Monotone: increases monotonically
Extremes: No
Convex:
at x< 0 : выпукла вниз
for x > 0 : convex up
Breakpoints: x=0, y=0
Intersection points with coordinate axes: x=0, y=0
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
;
Private values:
for x = -1, y(-1) = -1
for x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

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

The properties of the power function y = x p with a rational exponent , being within 0 are presented.< p < 1 , где n = 2, 4, 6, ... - четное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

Domain: -∞ < x < +∞
Multiple values: 0 ≤ y< +∞
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 : монотонно убывает
for x > 0 : monotonically increasing
Extremes: minimum at x = 0, y = 0
Convex: convex upward at x ≠ 0
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Sign: for x ≠ 0, y > 0
Limits:
;
Private values:
for x = -1, y(-1) = 1
for x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

The exponent 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, ... is odd.

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

Properties of a power function y = x p with a rational exponent greater than one: . Where n = 5, 7, 9, ... is an odd natural number, m = 3, 5, 7 ... is an odd natural number.

Domain: -∞ < x < ∞
Multiple values: -∞ < y < ∞
Parity: odd, y(-x) = - y(x)
Monotone: increases monotonically
Extremes: No
Convex:
at -∞< x < 0 выпукла вверх
at 0< x < ∞ выпукла вниз
Breakpoints: x=0, y=0
Intersection points with coordinate axes: x=0, y=0
Limits:
;
Private values:
for x = -1, y(-1) = -1
for x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

Even numerator, n = 4, 6, 8, ...

Properties of a power function y = x p with a rational exponent greater than one: . Where n = 4, 6, 8, ... is an even natural number, m = 3, 5, 7 ... is an odd natural number.

Domain: -∞ < x < ∞
Multiple values: 0 ≤ y< ∞
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 монотонно убывает
for x > 0 monotonically increases
Extremes: minimum at x = 0, y = 0
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Limits:
;
Private values:
for x = -1, y(-1) = 1
for x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

The denominator of the fractional indicator is even

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 those 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 considered above in that they are not defined for negative values ​​of the x argument. 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.


y = x p for different values ​​of the exponent p .

Power function with negative p< 0

Domain: x > 0
Multiple values: y > 0
Monotone: decreases monotonically
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: No
Limits: ;
private value: For x = 1, y(1) = 1 p = 1

Power function with positive exponent p > 0

The indicator is less than one 0< p < 1

Domain: x ≥ 0
Multiple values: y ≥ 0
Monotone: increases monotonically
Convex: convex up
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Limits:
Private values: For x = 0, y(0) = 0 p = 0 .
For x = 1, y(1) = 1 p = 1

The indicator is greater than one p > 1

Domain: x ≥ 0
Multiple values: y ≥ 0
Monotone: increases monotonically
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Limits:
Private values: For x = 0, y(0) = 0 p = 0 .
For x = 1, y(1) = 1 p = 1

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.

See also:

For the convenience of considering a power function, we will consider 4 separate cases: a power function with a natural exponent, a power function with an integer exponent, a power function with a rational exponent, and a power function with an irrational exponent.

Power function with natural exponent

To begin with, we introduce the concept of a degree with a natural exponent.

Definition 1

The power of a real number $a$ with natural exponent $n$ is a number equal to the product of $n$ factors, each of which is equal to the number $a$.

Picture 1.

$a$ is the base of the degree.

$n$ - exponent.

Consider now a power function with a natural exponent, its properties and graph.

Definition 2

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

For further convenience, consider separately the power function with even exponent $f\left(x\right)=x^(2n)$ and the power function with odd exponent $f\left(x\right)=x^(2n-1)$ ($n\in N)$.

Properties of a power function with natural even exponent

    $f\left(-x\right)=((-x))^(2n)=x^(2n)=f(x)$ is an even function.

    Scope -- $ \

    The function decreases as $x\in (-\infty ,0)$ and increases as $x\in (0,+\infty)$.

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

    The function is convex on the entire domain of definition.

    Behavior at the ends of the scope:

    \[(\mathop(lim)_(x\to -\infty ) x^(2n)\ )=+\infty \] \[(\mathop(lim)_(x\to +\infty ) x^( 2n)\ )=+\infty \]

    Graph (Fig. 2).

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

Properties of a power function with natural odd exponent

    The domain of definition is all real numbers.

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

    $f(x)$ is continuous on 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

To begin with, we introduce the concept of a degree with an integer exponent.

Definition 3

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

Figure 4

Consider now 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 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 considered it above. For $n=0$ we get a linear function $y=1$. We 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 scope 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 on the entire domain of definition.

    Range of value:

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

    If the exponent is odd, 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

A power function is a function of the form y=x n (read as y equals x to the power of n), where n is some given number. Particular cases of power functions are functions of the form y=x, y=x 2 , y=x 3 , y=1/x and many others. Let's talk more about each of them.

Linear function y=x 1 (y=x)

The graph is a straight line passing through the point (0; 0) at an angle of 45 degrees to the positive direction of the Ox axis.

The chart is shown below.

Basic properties of a linear function:

  • The function is increasing and is defined on the whole number axis.
  • It has no maximum and minimum values.

Quadratic function y=x 2

The graph of a quadratic function is a parabola.

Basic properties of a quadratic function:

  • 1. For x=0, y=0, and y>0 for x0
  • 2. The quadratic function reaches its minimum value at its vertex. Ymin at x=0; It should also be noted that the maximum value of the function does not exist.
  • 3. The function decreases on the interval (-∞; 0] and increases on the interval )
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