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 for various values of the exponent n = 1, 3, 5, ....
Domain: -∞ < x < ∞
Multiple meanings: -∞ < y < ∞
Parity: odd, y(-x) = - y(x)
Monotone: monotonically increases
Extremes: No
Convex:
at -∞< x < 0
выпукла вверх
at 0< x < ∞
выпукла вниз
Inflection points: x = 0, y = 0
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, ... . 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.
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 meanings: 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
Inflection points: No
Intersection points with coordinate axes: x = 0, y = 0
Limits:
;
Private values:
at x = -1, y(-1) = (-1) n ≡ (-1) 2k = 1
at 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 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:
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 meanings: y ≠ 0
Parity: odd, y(-x) = - y(x)
Monotone: monotonically decreases
Extremes: No
Convex:
at x< 0
:
выпукла вверх
for x > 0: convex downward
Inflection points: 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:
when n = -1,
at 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 meanings: y > 0
Parity: even, y(-x) = y(x)
Monotone:
at x< 0
:
монотонно возрастает
for x > 0: monotonically decreases
Extremes: No
Convex: convex down
Inflection points: No
Intersection points with coordinate axes: No
Sign: y > 0
Limits:
; ; ;
Private values:
for x = 1, y(1) = 1 n = 1
Reverse function:
at n = -2,
at 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 values of the argument x. Let us consider the properties of such power functions when the exponent p is within certain limits.
The p-value is negative, p< 0
Let the rational exponent (with odd denominator m = 3, 5, 7, ...) be less than zero: .
Graphs of power 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, ...
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.
Domain: x ≠ 0
Multiple meanings: y ≠ 0
Parity: odd, y(-x) = - y(x)
Monotone: monotonically decreases
Extremes: No
Convex:
at x< 0
:
выпукла вверх
for x > 0: convex downward
Inflection points: No
Intersection points with coordinate axes: No
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
; ; ;
Private values:
at 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 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.
Domain: x ≠ 0
Multiple meanings: y > 0
Parity: even, y(-x) = y(x)
Monotone:
at x< 0
:
монотонно возрастает
for x > 0: monotonically decreases
Extremes: No
Convex: convex down
Inflection points: No
Intersection points with coordinate axes: No
Sign: y > 0
Limits:
; ; ;
Private values:
at 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 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 meanings: -∞ < y < +∞
Parity: odd, y(-x) = - y(x)
Monotone: monotonically increases
Extremes: No
Convex:
at x< 0
:
выпукла вниз
for x > 0: convex upward
Inflection points: 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:
at x = -1, y(-1) = -1
at 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 within 0 are presented< p < 1 , где n = 2, 4, 6, ... - четное натуральное, m = 3, 5, 7 ... - нечетное натуральное.
Domain: -∞ < x < +∞
Multiple meanings: 0 ≤ y< +∞
Parity: even, y(-x) = y(x)
Monotone:
at x< 0
:
монотонно убывает
for x > 0: increases monotonically
Extremes: minimum at x = 0, y = 0
Convex: convex upward for x ≠ 0
Inflection points: No
Intersection points with coordinate axes: x = 0, y = 0
Sign: for x ≠ 0, y > 0
Limits:
;
Private values:
at x = -1, y(-1) = 1
at x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:
The p index 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, ...
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.
Domain: -∞ < x < ∞
Multiple meanings: -∞ < y < ∞
Parity: odd, y(-x) = - y(x)
Monotone: monotonically increases
Extremes: No
Convex:
at -∞< x < 0
выпукла вверх
at 0< x < ∞
выпукла вниз
Inflection points: x = 0, y = 0
Intersection points with coordinate axes: x = 0, y = 0
Limits:
;
Private values:
at x = -1, y(-1) = -1
at x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:
Even numerator, n = 4, 6, 8, ...
Properties of the power function y = x p with a rational exponent greater than one: . Where n = 4, 6, 8, ... - even natural, m = 3, 5, 7 ... - odd natural.
Domain: -∞ < x < ∞
Multiple meanings: 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
Inflection points: No
Intersection points with coordinate axes: x = 0, y = 0
Limits:
;
Private values:
at x = -1, y(-1) = 1
at 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 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.
y = x p for different values of the exponent p.
Power function with negative exponent p< 0
Domain: x > 0
Multiple meanings: y > 0
Monotone: monotonically decreases
Convex: convex down
Inflection points: No
Intersection points with coordinate axes: No
Limits: ;
Private meaning: For x = 1, y(1) = 1 p = 1
Power function with positive exponent p > 0
Indicator less than one 0< p < 1
Domain: x ≥ 0
Multiple meanings: y ≥ 0
Monotone: monotonically increases
Convex: convex upward
Inflection points: 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 meanings: y ≥ 0
Monotone: monotonically increases
Convex: convex down
Inflection points: 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 college students, “Lan”, 2009.
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Slide captions:
Lesson topic: Power function and its graph.
Just as algebraists write A 2, A 3, ... instead of AA, AAA, ..., so I write instead of a -1, a -2, a -3, ... Newton I.
y = x x y y = x 2 x y y = x 3 x y x y Direct Parabola Cubic parabola Hyperbola We are familiar with the functions: All these functions are special cases of the power function
where p is a given real number Definition: A power function is a function of the form y = x p. The properties and graph of a power function depend on the properties of a power with a real exponent, and in particular on the values of x and p for which the power x p makes sense.
The function y=x 2 n is even, because (– x) 2 n = x 2 n Function decreases on the interval Function increases on the interval Power function: Exponent p = 2n – even natural number y = x 2, y = x 4, y = x 6, y = x 8, ... 1 0 x y y = x 2
y x - 1 0 1 2 y = x 2 y = x 6 y = x 4 Power function: Exponent p = 2n – even natural number y = x 2, y = x 4, y = x 6, y = x 8, ...
The function y=x 2 n -1 is odd, because (– x) 2 n -1 = – x 2 n -1 Function increases on the interval Power function: Exponent p = 2n-1 – odd natural number y = x 3, y = x 5, y = x 7, y = x 9 , … 1 0
Power function: y x - 1 0 1 2 y = x 3 y = x 7 y = x 5 Exponent p = 2n-1 – odd natural number y = x 3, y = x 5, y = x 7, y = x 9 ,...
The function y=x- 2 n is even, because (– x) -2 n = x -2 n The function increases on the interval The function decreases on the interval Power function: Exponent p = -2n – where n is a natural number y = x -2, y = x -4, y = x -6 , y = x -8 , … 0 1
1 0 1 2 y = x -4 y = x -2 y = x -6 Power function: Exponent p = -2n – where n is a natural number y = x -2, y = x -4, y = x -6, y = x -8, ... y x
The function decreases on the interval The function y=x -(2 n -1) is odd, because (– x) –(2 n -1) = – x –(2 n -1) The function decreases on the interval Power function: Exponent p = -(2n-1) – where n is a natural number y = x -3, y = x -5, y = x -7, y = x -9, ... 1 0
y = x -1 y = x -3 y = x -5 Power function: Exponent p = -(2n-1) – where n is a natural number y = x -3, y = x -5, y = x -7, y = x -9 , … y x - 1 0 1 2
Power function: Exponent p – positive real non-integer number y = x 1.3, y = x 0.7, y = x 2.2, y = x 1/3,… 0 1 x y The function increases over the interval
y = x 0.7 Power function: Exponent p – positive real non-integer number y = x 1.3, y = x 0.7, y = x 2.2, y = x 1/3,… y x - 1 0 1 2 y = x 0.5 y = x 0.84
Power function: Exponent p – positive real non-integer number y = x 1.3, y = x 0.7, y = x 2.2, y = x 1/3,… y x - 1 0 1 2 y = x 1, 5 y = x 3.1 y = x 2.5
Power function: Exponent p – negative real non-integer number y= x -1.3, y= x -0.7, y= x -2.2, y = x -1/3,… 0 1 x y The function decreases by in between
y = x -0.3 y = x -2.3 y = x -3.8 Power function: Exponent p – negative real non-integer number y= x -1.3, y= x -0.7, y= x -2.2, y = x -1/3,… y x - 1 0 1 2 y = x -1.3
On the topic: methodological developments, presentations and notes
The use of integration in the educational process as a way to develop analytical and creative abilities....
Provides reference data on the exponential function - basic properties, graphs and formulas. The following topics are considered: domain of definition, set of values, monotonicity, inverse function, derivative, integral, power series expansion and representation using complex numbers.
ContentProperties of the Exponential Function
The exponential function y = a x has the following properties on the set of real numbers ():
(1.1)
defined and continuous, for , for all ;
(1.2)
for a ≠ 1
has many meanings;
(1.3)
strictly increases at , strictly decreases at ,
is constant at ;
(1.4)
at ;
at ;
(1.5)
;
(1.6)
;
(1.7)
;
(1.8)
;
(1.9)
;
(1.10)
;
(1.11)
,
.
Other useful formulas.
.
Formula for converting to an exponential function with a different exponent base:
When b = e, we obtain the expression of the exponential function through the exponential:
Private values
, , , , .
y = a x for different values of the base a. The figure shows graphs of the exponential function
y (x) = ax
for four values degree bases: a = 2
, a = 8
, a = 1/2
and a = 1/8
. It can be seen that for a > 1
the exponential function increases monotonically. The larger the base of the degree a, the stronger the growth. At 0
< a < 1
the exponential function decreases monotonically. The smaller the exponent a, the stronger the decrease.
Ascending, descending
The exponential function for is strictly monotonic and therefore has no extrema. Its main properties are presented in the table.
| y = a x , a > 1 | y = ax, 0 < a < 1 | |
| Domain | - ∞ < x < + ∞ | - ∞ < x < + ∞ |
| Range of values | 0 < y < + ∞ | 0 < y < + ∞ |
| Monotone | monotonically increases | monotonically decreases |
| Zeros, y = 0 | No | No |
| Intercept points with the ordinate axis, x = 0 | y = 1 | y = 1 |
| + ∞ | 0 | |
| 0 | + ∞ |
Inverse function
The inverse of an exponential function with base a is the logarithm to base a.
If , then
.
If , then
.
Differentiation of an exponential function
To differentiate an exponential function, its base must be reduced to the number e, apply the table of derivatives and the rule for differentiating a complex function.
To do this you need to use the property of logarithms
and the formula from the derivatives table:
.
Let an exponential function be given:
.
We bring it to the base e:
Let's apply the rule of differentiation of complex functions. To do this, introduce the variable
Then
From the table of derivatives we have (replace the variable x with z):
.
Since is a constant, the derivative of z with respect to x is equal to
.
According to the rule of differentiation of a complex function:
.
Derivative of an exponential function
.
Derivative of nth order:
.
Deriving formulas > > >
An example of differentiating an exponential function
Find the derivative of a function
y = 3 5 x
Solution
Let's express the base of the exponential function through the number e.
3 = e ln 3
Then
.
Enter a variable
.
Then
From the table of derivatives we find:
.
Because the 5ln 3 is a constant, then the derivative of z with respect to x is equal to:
.
According to the rule of differentiation of a complex function, we have:
.
Answer
Integral
Expressions using complex numbers
Consider the complex number function z:
f (z) = a z
where z = x + iy; i 2 = - 1
.
Let us express the complex constant a in terms of modulus r and argument φ:
a = r e i φ
Then
.
The argument φ is not uniquely defined. In general
φ = φ 0 + 2 πn,
where n is an integer. Therefore the function f (z) is also not clear. Its main significance is often considered
.
Are you familiar with the functions 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 significantly depend on the properties of a power with a real exponent, and in particular on the values for which x And p degree makes sense x p. Let us proceed to a similar consideration of various cases depending on
exponent p.
- Index p=2n-even natural number.
properties:
- domain of definition - 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.
2. Indicator p=2n-1- odd natural number
In this case, the power function y=x2n-1, where is a natural number, has the following properties:
- domain of definition - set R;
- set of values - set R;
- function y=x2n-1 odd because (- x) 2n-1=x2n-1;
- the function is increasing on the entire real axis.
3.Indicator p=-2n, Where n- natural number.
In this case, the power function y=x -2n =1/x 2n has the following properties:
- domain of definition - set R, except x=0;
- set of values - positive numbers y>0;
- function y =1/x2n even, because 1/(-x)2n=1/x 2n;
- the function is increasing on the interval x<0 и убывающей на промежутке x>0.
