Properties of the function y 4 to the power of x. Basic elementary functions
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
First, let's 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 equals the number $a$.
Picture 1.
$a$ is the base of the degree.
$n$ is the exponent.
Let us now consider 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 a natural exponent.
For further convenience, we consider separately a power function with an even exponent $f\left(x\right)=x^(2n)$ and a power function with an odd exponent $f\left(x\right)=x^(2n-1)$ ($n\in N)$.
Properties of a power function with a natural even exponent
$f\left(-x\right)=((-x))^(2n)=x^(2n)=f(x)$ -- the function is even.
Value area -- $\
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 over the entire domain of definition.
Behavior at the ends of the domain:
\[(\mathop(lim)_(x\to -\infty ) x^(2n)\ )=+\infty \] \[(\mathop(lim)_(x\to +\infty ) x^( 2n)\ )=+\infty \]
Graph (Fig. 2).
Figure 2. Graph of the function $f\left(x\right)=x^(2n)$
Properties of a power function with a natural odd exponent
Scope -- 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 $x\in \left(-\infty ,0\right)(0,+\infty)$. If the exponent is even, 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
National Research University
Department of Applied Geology
Abstract on higher mathematics
On the topic: “Basic elementary functions,
their properties and graphs"
Completed:
Checked:
teacher
Definition. The function given by the formula y=a x (where a>0, a≠1) is called an exponential function with base a.
Let us formulate the main properties exponential function:
1. The domain of definition is the set (R) of all real numbers.
2. Range - the set (R+) of all positive real numbers.
3. For a > 1, the function increases along the entire number line; at 0<а<1 функция убывает.
4. Is a function of general form.
, on the interval xО [-3;3]
, on the interval xО [-3;3] A function of the form y(x)=x n, where n is the number ОR, is called a power function. The number n can take on different values: both integer and fractional, both even and odd. Depending on this, the power function will have a different form. Let's consider special cases that are power functions and reflect the basic properties of this type of curve in the following order: power function y=x² (function with an even exponent - a parabola), power function y=x³ (function with an odd exponent - cubic parabola) and function y=√x (x to the power of ½) (function with a fractional exponent), function with a negative integer exponent (hyperbola).
Power function y=x²
1. D(x)=R – the function is defined on the entire numerical axis;
2. E(y)= and increases on the interval
Power function y=x³
1. The graph of the function y=x³ is called a cubic parabola. The power function y=x³ has the following properties:
2. D(x)=R – the function is defined on the entire numerical axis;
3. E(y)=(-∞;∞) – the function takes all values in its domain of definition;
4. When x=0 y=0 – the function passes through the origin of coordinates O(0;0).
5. The function increases over the entire domain of definition.
6. The function is odd (symmetrical about the origin).
, on the interval xО [-3;3] Depending on the numerical factor in front of x³, the function can be steep/flat and increasing/decreasing.
Power function with negative integer exponent:
If the exponent n is odd, then the graph of such a power function is called a hyperbola. A power function with an integer negative exponent has the following properties:
1. D(x)=(-∞;0)U(0;∞) for any n;
2. E(y)=(-∞;0)U(0;∞), if n is an odd number; E(y)=(0;∞), if n is an even number;
3. The function decreases over the entire domain of definition if n is an odd number; the function increases on the interval (-∞;0) and decreases on the interval (0;∞) if n is an even number.
4. The function is odd (symmetrical about the origin) if n is an odd number; a function is even if n is an even number.
5. The function passes through the points (1;1) and (-1;-1) if n is an odd number and through the points (1;1) and (-1;1) if n is an even number.
, on the interval xО [-3;3] Power function with fractional exponent
A power function with a fractional exponent (picture) has a graph of the function shown in the figure. A power function with a fractional exponent has the following properties: (picture)
1. D(x) ОR, if n is an odd number and D(x)= 
, on the interval xО 
, on the interval xО [-3;3]
The logarithmic function y = log a x has the following properties:
1. Domain of definition D(x)О (0; + ∞).
2. Range of values E(y) О (- ∞; + ∞)
3. The function is neither even nor odd (of general form).
4. The function increases on the interval (0; + ∞) for a > 1, decreases on (0; + ∞) for 0< а < 1.
The graph of the function y = log a x can be obtained from the graph of the function y = a x using a symmetry transformation about the straight line y = x. Figure 9 shows a graph of the logarithmic function for a > 1, and Figure 10 for 0< a < 1.
; on the interval xО 
; on the interval xО The functions y = sin x, y = cos x, y = tan x, y = ctg x are called trigonometric functions.
The functions y = sin x, y = tan x, y = ctg x are odd, and the function y = cos x is even.
Function y = sin(x).
1. Domain of definition D(x) ОR.
2. Range of values E(y) О [ - 1; 1].
3. The function is periodic; the main period is 2π.
4. The function is odd.
5. The function increases on intervals [ -π/2 + 2πn; π/2 + 2πn] and decreases on the intervals [π/2 + 2πn; 3π/2 + 2πn], n О Z.
The graph of the function y = sin (x) is shown in Figure 11.
The methodological material is for reference only and applies to a wide range of topics. The article provides an overview of graphs of basic elementary functions and discusses the most important question – how to build a graph correctly and QUICKLY. During the study higher mathematics Without knowing the graphs of basic elementary functions, it will be difficult, so it is very important to remember what the graphs of a parabola, hyperbola, sine, cosine, etc. look like, and remember some of the function values. We will also talk about some properties of the main functions.
I do not claim completeness and scientific thoroughness of the materials; the emphasis will be placed, first of all, on practice - those things with which one encounters literally at every step, in any topic of higher mathematics. Charts for dummies? One could say so.
Due to numerous requests from readers clickable table of contents:
In addition, there is an ultra-short synopsis on the topic
– master 16 types of charts by studying SIX pages!
Seriously, six, even I was surprised. This summary contains improved graphics and is available for a nominal fee; a demo version can be viewed. It is convenient to print the file so that the graphs are always at hand. Thanks for supporting the project!
And let's start right away:
How to construct coordinate axes correctly?
In practice, tests are almost always completed by students in separate notebooks, lined in a square. Why do you need checkered markings? After all, the work, in principle, can be done on A4 sheets. And the cage is necessary just for high-quality and accurate design of drawings.
Any drawing of a function graph begins with coordinate axes.
Drawings can be two-dimensional or three-dimensional.
Let's first consider the two-dimensional case Cartesian rectangular coordinate system:
1) Draw coordinate axes. The axis is called x-axis , and the axis is y-axis . We always try to draw them neat and not crooked. The arrows should also not resemble Papa Carlo’s beard.
2) We sign the axes with large letters “X” and “Y”. Don't forget to label the axes.
3) Set the scale along the axes: draw a zero and two ones. When making a drawing, the most convenient and frequently used scale is: 1 unit = 2 cells (drawing on the left) - if possible, stick to it. However, from time to time it happens that the drawing does not fit on the notebook sheet - then we reduce the scale: 1 unit = 1 cell (drawing on the right). It’s rare, but it happens that the scale of the drawing has to be reduced (or increased) even more
There is NO NEED to “machine gun” …-5, -4, -3, -1, 0, 1, 2, 3, 4, 5, …. For the coordinate plane is not a monument to Descartes, and the student is not a dove. We put zero And two units along the axes. Sometimes instead of units, it is convenient to “mark” other values, for example, “two” on the abscissa axis and “three” on the ordinate axis - and this system (0, 2 and 3) will also uniquely define the coordinate grid.
It is better to estimate the estimated dimensions of the drawing BEFORE constructing the drawing. So, for example, if the task requires drawing a triangle with vertices , , , then it is completely clear that the popular scale of 1 unit = 2 cells will not work. Why? Let's look at the point - here you will have to measure fifteen centimeters down, and, obviously, the drawing will not fit (or barely fit) on a notebook sheet. Therefore, we immediately select a smaller scale: 1 unit = 1 cell.
By the way, about centimeters and notebook cells. Is it true that 30 notebook cells contain 15 centimeters? For fun, measure 15 centimeters in your notebook with a ruler. In the USSR, this may have been true... It is interesting to note that if you measure these same centimeters horizontally and vertically, the results (in the cells) will be different! Strictly speaking, modern notebooks are not checkered, but rectangular. This may seem nonsense, but drawing, for example, a circle with a compass in such situations is very inconvenient. To be honest, at such moments you begin to think about the correctness of Comrade Stalin, who was sent to camps for hack work in production, not to mention the domestic automobile industry, falling planes or exploding power plants.
Speaking of quality, or brief recommendation for stationery. Today, most of the notebooks on sale are, to say the least, complete crap. For the reason that they get wet, and not only from gel pens, but also from ballpoint pens! They save money on paper. For registration tests I recommend using notebooks from the Arkhangelsk Pulp and Paper Mill (18 sheets, grid) or “Pyaterochka”, although it is more expensive. It is advisable to choose a gel pen; even the cheapest Chinese gel refill is much better than a ballpoint pen, which either smudges or tears the paper. The only "competitive" ballpoint pen in my memory is "Erich Krause". She writes clearly, beautifully and consistently – whether with a full core or with an almost empty one.
Additionally: The vision of a rectangular coordinate system through the eyes of analytical geometry is covered in the article Linear (non) dependence of vectors. Basis of vectors, detailed information about coordinate quarters can be found in the second paragraph of the lesson Linear inequalities.
3D case
It's almost the same here.
1) Draw coordinate axes. Standard: axis applicate – directed upwards, axis – directed to the right, axis – directed downwards to the left strictly at an angle of 45 degrees.
2) Label the axes.
3) Set the scale along the axes. The scale along the axis is two times smaller than the scale along the other axes. Also note that in the right drawing I used a non-standard "notch" along the axis (this possibility has already been mentioned above). From my point of view, this is more accurate, faster and more aesthetically pleasing - there is no need to look for the middle of the cell under a microscope and “sculpt” a unit close to the origin of coordinates.
When making a 3D drawing, again, give priority to scale
1 unit = 2 cells (drawing on the left).
What are all these rules for? Rules are made to be broken. That's what I'll do now. The fact is that subsequent drawings of the article will be made by me in Excel, and the coordinate axes will look incorrect from the point of view of correct design. I could draw all the graphs by hand, but it’s actually scary to draw them as Excel is reluctant to draw them much more accurately.
Graphs and basic properties of elementary functions
Linear function is given by the equation. The graph of linear functions is direct. In order to construct a straight line, it is enough to know two points.
Example 1
Construct a graph of the function. Let's find two points. It is advantageous to choose zero as one of the points.
If , then
Let's take another point, for example, 1.
If , then
When completing tasks, the coordinates of the points are usually summarized in a table:
And the values themselves are calculated orally or on a draft, a calculator.
Two points have been found, let’s make the drawing:
When preparing a drawing, we always sign the graphics.
It would be useful to recall special cases of a linear function:
Notice how I placed the signatures, signatures should not allow discrepancies when studying the drawing. IN in this case It was extremely undesirable to put a signature next to the point of intersection of the lines, or at the bottom right between the graphs.
1) A linear function of the form () is called direct proportionality. For example, . A direct proportionality graph always passes through the origin. Thus, constructing a straight line is simplified - it is enough to find just one point.
2) An equation of the form specifies a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is constructed immediately, without finding any points. That is, the entry should be understood as follows: “the y is always equal to –4, for any value of x.”
3) An equation of the form specifies a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is also plotted immediately. The entry should be understood as follows: “x is always, for any value of y, equal to 1.”
Some will ask, why remember 6th grade?! That’s how it is, maybe it’s so, but over the years of practice I’ve met a good dozen students who were baffled by the task of constructing a graph like or.
Constructing a straight line is the most common action when making drawings.
The straight line is discussed in detail in the course of analytical geometry, and those interested can refer to the article Equation of a straight line on a plane.
Graph of a quadratic, cubic function, graph of a polynomial
Parabola. Graph of a quadratic function 
() represents a parabola. Consider the famous case:
Let's recall some properties of the function.
So, the solution to our equation: – it is at this point that the vertex of the parabola is located. Why this is so can be found in the theoretical article on the derivative and the lesson on extrema of the function. In the meantime, let’s calculate the corresponding “Y” value:
Thus, the vertex is at the point
Now we find other points, while brazenly using the symmetry of the parabola. It should be noted that the function 
– is not even, but, nevertheless, no one canceled the symmetry of the parabola.
In what order to find the remaining points, I think it will be clear from the final table:
This construction algorithm can figuratively be called a “shuttle” or the “back and forth” principle with Anfisa Chekhova.
Let's make the drawing:
From the graphs examined, another useful feature comes to mind:
For a quadratic function 
() the following is true:
If , then the branches of the parabola are directed upward.
If , then the branches of the parabola are directed downward.
In-depth knowledge about the curve can be obtained in the lesson Hyperbola and parabola.
A cubic parabola is given by the function. Here is a drawing familiar from school:
Let us list the main properties of the function
Graph of a function
It represents one of the branches of a parabola. Let's make the drawing:
Main properties of the function:
In this case, the axis is vertical asymptote for the graph of a hyperbola at .
Will BIG mistake, if, when drawing up a drawing, you carelessly allow the graph to intersect with an asymptote.
Also one-sided limits tell us that the hyperbola not limited from above And not limited from below.
Let’s examine the function at infinity: , that is, if we start moving along the axis to the left (or right) to infinity, then the “games” will be in an orderly step infinitely close approach zero, and, accordingly, the branches of the hyperbola infinitely close approach the axis.
So the axis is horizontal asymptote for the graph of a function, if “x” tends to plus or minus infinity.
The function is odd, and, therefore, the hyperbola is symmetrical about the origin. This fact obvious from the drawing, in addition, it is easily verified analytically: 
.
The graph of a function of the form () represents two branches of a hyperbola.
If , then the hyperbola is located in the first and third coordinate quarters(see picture above).
If , then the hyperbola is located in the second and fourth coordinate quarters.
The indicated pattern of hyperbola residence is easy to analyze from the point of view of geometric transformations of graphs.
Example 3
Construct the right branch of the hyperbola
We use the point-wise construction method, and it is advantageous to select the values so that they are divisible by a whole:
Let's make the drawing:
It will not be difficult to construct the left branch of the hyperbola; the oddness of the function will help here. Roughly speaking, in the table of pointwise construction, we mentally add a minus to each number, put the corresponding points and draw the second branch.
Detailed geometric information about the line considered can be found in the article Hyperbola and parabola.
Graph of an Exponential Function
In this section, I will immediately consider the exponential function, since in problems of higher mathematics in 95% of cases it is the exponential that appears.
I remind you that this is irrational number: , this will be required when constructing a graph, which, in fact, I will build without ceremony. Three points are probably enough:
Let's leave the graph of the function alone for now, more on it later.
Main properties of the function:
Function graphs, etc., look fundamentally the same.
I must say that the second case occurs less frequently in practice, but it does occur, so I considered it necessary to include it in this article.
Graph of a logarithmic function
Consider a function with natural logarithm.
Let's make a point-by-point drawing:
If you have forgotten what a logarithm is, please refer to your school textbooks.
Main properties of the function:
Domain: 
Range of values: .
The function is not limited from above: 
, albeit slowly, but the branch of the logarithm goes up to infinity.
Let us examine the behavior of the function near zero on the right: 
. So the axis is vertical asymptote
for the graph of a function as “x” tends to zero from the right.
It is imperative to know and remember the typical value of the logarithm: .
The graph of the logarithm at the base looks fundamentally the same: , , ( decimal logarithm to base 10), etc. Moreover, the larger the base, the flatter the graph will be.
We will not consider the case, I don’t remember when last time I built a graph on this basis. And the logarithm seems to be a very rare guest in problems of higher mathematics.
At the end of this paragraph I will say one more fact: Exponential function and logarithmic function– these are two mutually inverse functions. If you look closely at the graph of the logarithm, you can see that this is the same exponent, it’s just located a little differently.
Graphs of trigonometric functions
Where does trigonometric torment begin at school? Right. From sine
Let's plot the function
This line called sinusoid.
Let me remind you that “pi” is an irrational number: , and in trigonometry it makes your eyes dazzle.
Main properties of the function:
This function is periodic with period . What does it mean? Let's look at the segment. To the left and right of it, exactly the same piece of the graph is repeated endlessly.
Domain: , that is, for any value of “x” there is a sine value.
Range of values: . The function is limited: , that is, all the “games” sit strictly in the segment .
This does not happen: or, more precisely, it happens, but these equations do not have a solution.
Power function, its properties and graph Demonstration material Lesson-lecture Concept of function. Function properties. Power function, its properties and graph. Grade 10 All rights reserved. Copyright with Copyright with
Lesson progress: Repetition. Function. Properties of functions. Learning new material. 1. Definition of a power function.Definition of a power function. 2. Properties and graphs of power functions. Properties and graphs of power functions. Consolidation of the studied material. Verbal counting. Verbal counting. Lesson summary. Homework assignment. Homework assignment.
Domain of definition and domain of values of a function All values of the independent variable form the domain of definition of the function x y=f(x) f Domain of definition of the function Domain of values of the function All values that the dependent variable takes form the domain of values of the function Function. Function properties
Graph of a function Let a function be given where xY y x.75 3 0.6 4 0.5 The graph of a function is the set of all points of the coordinate plane, the abscissas of which are equal to the values of the argument, and the ordinates are equal to the corresponding values of the function. Function. Function properties
Y x Domain of definition and range of values of the function 4 y=f(x) Domain of definition of the function: Domain of values of the function: Function. Function properties
Even function y x y=f(x) The graph of an even function is symmetrical with respect to the axis of the op-amp. The function y=f(x) is called even if f(-x) = f(x) for any x from the domain of definition of the function Function. Function properties
Odd function y x y=f(x) The graph of an odd function is symmetrical with respect to the origin O(0;0) The function y=f(x) is called odd if f(-x) = -f(x) for any x from the region function definitions Function. Function properties
Definition of a power function A function where p is a given real number is called a power function. p y=x p P=x y 0 Lesson progress
Power function x y 1. The domain of definition and the range of values of power functions of the form, where n – natural number, are all real numbers. 2. These functions are odd. Their graph is symmetrical about the origin. Properties and graphs of power functions
Power functions with rational positive exponent Domain - all positive numbers and the number 0. The range of values of functions with this exponent is also all positive numbers and the number 0. These functions are neither even nor odd. y x Properties and graphs of power functions
Power function with rational negative exponent. The domain of definition and range of values of such functions are all positive numbers. The functions are neither even nor odd. Such functions decrease throughout their entire domain of definition. y x Properties and graphs of power functions Lesson progress
The properties and graphs of power functions are presented for different meanings exponent. Basic formulas, domains of definition and sets of values, parity, monotonicity, increasing and decreasing, extrema, convexity, inflections, points of intersection with coordinate axes, limits, particular values.
Formulas with power functions
On the domain of definition of the power function y = x p we have following formulas:
;
;
;
;
;
;
;
;
.
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, 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 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, ... - 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 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.
