Various functions and their graphs. Function properties. Graphical way to specify a function
This teaching material is for reference only and relates to a wide range of topics. The article provides an overview of graphs of basic elementary functions and considers the most important issue - how to build a graph correctly and QUICKLY. In the course of studying higher mathematics without knowledge of 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 meanings of the functions. 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 a super-short summary 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 a brief recommendation on 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. To complete tests, I recommend using notebooks from the Arkhangelsk Pulp and Paper Mill (18 sheets, square) 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 I can remember is the 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
A 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 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 plotted 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 .
It would be a GROSS 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 to move along the axis to the left (or right) to infinity, then the “games” will step smoothly 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 is 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.
Let me remind you that this is an 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 a 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: .
In principle, the graph of the logarithm to the base looks the same: , , (decimal logarithm to the base 10), etc. Moreover, the larger the base, the flatter the graph will be.
We won’t consider the case; I don’t remember the last time I built a graph with such a 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 is 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 “players” sit strictly in the segment .
This does not happen: or, more precisely, it happens, but these equations do not have a solution.
Function is one of the most important mathematical concepts. Function - variable dependency at from variable x, if each value X matches a single value at. Variable X called the independent variable or argument. Variable at called the dependent variable. All values of the independent variable (variable x) form the domain of definition of the function. All values that the dependent variable takes (variable y), form the range of values of the function.
Function graph call 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, that is, the values of the variable are plotted along the abscissa axis x, and the values of the variable are plotted along the ordinate axis y. To do this, you need to know the properties of the function. The main properties of the function will be discussed below!
To plot a function graph, we recommend using our program -. If you have any questions while studying the material on this page, you can always ask them on ours. Also on the forum they will help you solve problems in mathematics, chemistry, and many other subjects!
Basic properties of functions.
1) Function domain and function range.
The domain of a function is the set of all valid valid argument values x(variable x), for which the function y = f(x) determined.
The range of a function is the set of all real values y, which the function accepts.
In elementary mathematics, functions are studied only on the set of real numbers.
2) Function zeros.
Function zero is the value of the argument at which the value of the function is equal to zero.
3) Intervals of constant sign of a function.
Intervals of constant sign of a function are sets of argument values on which the function values are only positive or only negative.
4) Monotonicity of the function.
An increasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a larger value of the function.
A decreasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.
5) Even (odd) function.
An even function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality f(-x) = f(x). The graph of an even function is symmetrical about the ordinate.
An odd function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality is true f(-x) = - f(x). The graph of an odd function is symmetrical about the origin.
6) Limited and unlimited functions.
A function is called bounded if there is a positive number M such that |f(x)| ≤ M for all values of x. If such a number does not exist, then the function is unlimited.
7) Periodicity of the function.
A function f(x) is periodic if there is a non-zero number T such that for any x from the domain of definition of the function the following holds: f(x+T) = f(x). This smallest number is called the period of the function. All trigonometric functions are periodic. (
Knowledge basic elementary functions, their properties and graphs no less important than knowing the multiplication tables. They are like the foundation, everything is based on them, everything is built from them and everything comes down to them.
In this article we will list all the main elementary functions, provide their graphs and give without conclusion or proof properties of basic elementary functions according to the scheme:
- behavior of a function at the boundaries of the domain of definition, vertical asymptotes (if necessary, see the article classification of discontinuity points of a function);
- even and odd;
- intervals of convexity (convexity upward) and concavity (convexity downward), inflection points (if necessary, see the article convexity of a function, direction of convexity, inflection points, conditions of convexity and inflection);
- oblique and horizontal asymptotes;
- singular points of functions;
- special properties of some functions (for example, the smallest positive period of trigonometric functions).
If you are interested in or, then you can go to these sections of the theory.
Basic elementary functions are: constant function (constant), nth root, power function, exponential, logarithmic function, trigonometric and inverse trigonometric functions.
Page navigation.
Permanent function.
A constant function is defined on the set of all real numbers by the formula , where C is some real number. A constant function associates each real value of the independent variable x with the same value of the dependent variable y - the value C. A constant function is also called a constant.
The graph of a constant function is a straight line parallel to the x-axis and passing through the point with coordinates (0,C). For example, let's show graphs of constant functions y=5, y=-2 and, which in the figure below correspond to the black, red and blue lines, respectively.
Properties of a constant function.
- Domain: the entire set of real numbers.
- The constant function is even.
- Range of values: a set consisting of the singular number C.
- A constant function is non-increasing and non-decreasing (that’s why it’s constant).
- It makes no sense to talk about convexity and concavity of a constant.
- There are no asymptotes.
- The function passes through the point (0,C) of the coordinate plane.
Root of the nth degree.
Let's consider the basic elementary function, which is given by the formula , where n is a natural number greater than one.
Root of the nth degree, n is an even number.
Let's start with the nth root function for even values of the root exponent n.
As an example, here is a picture with images of function graphs 
and , they correspond to black, red and blue lines.
The graphs of even-degree root functions have a similar appearance for other values of the exponent.
Properties of the nth root function for even n.
The nth root, n is an odd number.
The nth root function with an odd root exponent n is defined on the entire set of real numbers. For example, here are the function graphs 
and , they correspond to black, red and blue curves.
For other odd values of the root exponent, the function graphs will have a similar appearance.
Properties of the nth root function for odd n.
Power function.
The power function is given by a formula of the form .
Let's consider the form of graphs of a power function and the properties of a power function depending on the value of the exponent.
Let's start with a power function with an integer exponent a. In this case, the appearance of graphs of power functions and the properties of the functions depend on the evenness or oddness of the exponent, as well as on its sign. Therefore, we will first consider power functions for odd positive values of the exponent a, then for even positive exponents, then for odd negative exponents, and finally, for even negative a.
The properties of power functions with fractional and irrational exponents (as well as the type of graphs of such power functions) depend on the value of the exponent a. We will consider them, firstly, for a from zero to one, secondly, for a greater than one, thirdly, for a from minus one to zero, fourthly, for a less than minus one.
At the end of this section, for completeness, we will describe a power function with zero exponent.
Power function with odd positive exponent.
Let's consider a power function with an odd positive exponent, that is, with a = 1,3,5,....
The figure below shows graphs of power functions – black line, – blue line, – red line, – green line. For a=1 we have linear function y=x.
Properties of a power function with an odd positive exponent.
Power function with even positive exponent.
Let's consider a power function with an even positive exponent, that is, for a = 2,4,6,....
As an example, we give graphs of power functions – black line, – blue line, – red line. For a=2 we have a quadratic function, the graph of which is quadratic parabola.
Properties of a power function with an even positive exponent.
Power function with odd negative exponent.
Look at the graphs of the power function for odd negative values of the exponent, that is, for a = -1, -3, -5,....
The figure shows graphs of power functions as examples - black line, - blue line, - red line, - green line. For a=-1 we have inverse proportionality, whose graph is hyperbola.
Properties of a power function with an odd negative exponent.
Power function with even negative exponent.
Let's move on to the power function at a=-2,-4,-6,….
The figure shows graphs of power functions – black line, – blue line, – red line.
Properties of a power function with an even negative exponent.
A power function with a rational or irrational exponent whose value is greater than zero and less than one.
Note! If a is a positive fraction with an odd denominator, then some authors consider the domain of definition of the power function to be the interval. It is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and principles of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values of the argument. We will adhere to precisely this view, that is, we will consider the set to be the domains of definition of power functions with fractional positive exponents. We recommend that students find out your teacher's opinion on this subtle point in order to avoid disagreements.
Let us consider a power function with a rational or irrational exponent a, and .
Let us present graphs of power functions for a=11/12 (black line), a=5/7 (red line), (blue line), a=2/5 (green line).
A power function with a non-integer rational or irrational exponent greater than one.
Let us consider a power function with a non-integer rational or irrational exponent a, and .
Let us present graphs of power functions given by the formulas 
(black, red, blue and green lines respectively).
For other values of the exponent a, the graphs of the function will have a similar appearance.
Properties of the power function at .
A power function with a real exponent that is greater than minus one and less than zero.
Note! If a is a negative fraction with an odd denominator, then some authors consider the domain of definition of a power function to be the interval 
. It is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and principles of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values of the argument. We will adhere to precisely this view, that is, we will consider the domains of definition of power functions with fractional fractional negative exponents to be a set, respectively. We recommend that students find out your teacher's opinion on this subtle point in order to avoid disagreements.
Let's move on to the power function, kgod.
To have a good idea of the form of graphs of power functions for , we give examples of graphs of functions 
(black, red, blue and green curves, respectively).
Properties of a power function with exponent a, .
A power function with a non-integer real exponent that is less than minus one.
Let us give examples of graphs of power functions for 
, they are depicted by black, red, blue and green lines, respectively.
Properties of a power function with a non-integer negative exponent less than minus one.
When a = 0, we have a function - this is a straight line from which the point (0;1) is excluded (it was agreed not to attach any significance to the expression 0 0).
Exponential function.
One of the main elementary functions is the exponential function.
The graph of the exponential function, where and takes different forms depending on the value of the base a. Let's figure this out.
First, consider the case when the base of the exponential function takes a value from zero to one, that is, .
As an example, we present graphs of the exponential function for a = 1/2 – blue line, a = 5/6 – red line. The graphs of the exponential function have a similar appearance for other values of the base from the interval.
Properties of an exponential function with a base less than one.
Let us move on to the case when the base of the exponential function is greater than one, that is, .
As an illustration, we present graphs of exponential functions - blue line and - red line. For other values of the base greater than one, the graphs of the exponential function will have a similar appearance.
Properties of an exponential function with a base greater than one.
Logarithmic function.
The next basic elementary function is the logarithmic function, where , . The logarithmic function is defined only for positive values of the argument, that is, for .
The graph of a logarithmic function takes different forms depending on the value of the base a.
What do the words mean? "set a function"? They mean: explain to everyone who wants to know what specific function we are talking. Moreover, explain clearly and unambiguously!
How can I do that? How set a function?
You can write a formula. You can draw a graph. You can make a table. Any way is some rule by which we can find out the value of the i for the x value we have chosen. Those. "set function", this means to show the law, the rule by which an x turns into a y.
Usually, in a variety of tasks there are already ready functions. They give us have already been set. Decide for yourself, yes, decide.) But... Most often, schoolchildren (and even students) work with formulas. They get used to it, you know... They get so used to it that any elementary question related to a different way of specifying a function immediately upsets the person...)
To avoid such cases, it makes sense to understand different ways of specifying functions. And, of course, apply this knowledge to “tricky” questions. It's quite simple. If you know what a function is...)
Go?)
Analytical method of specifying a function.
The most universal and powerful way. A function defined analytically this is the function that is given formulas. Actually, this is the whole explanation.) Functions that are familiar to everyone (I want to believe!), for example: y = 2x, or y = x 2 etc. and so on. are specified analytically.
By the way, not every formula can define a function. Not every formula meets the strict condition from the definition of a function. Namely - for every X there can only be one igrek. For example, in the formula y = ±x, For one values x=2, it turns out two y values: +2 and -2. This formula cannot define a unique function. As a rule, they don’t work with multi-valued functions in this branch of mathematics, in calculus.
What is good about the analytical way of specifying a function? Because if you have a formula, you know about the function All! You can make a sign. Build a graph. Explore this feature in full. Predict exactly where and how this function will behave. All mathematical analysis is based on this method of specifying functions. Let's say, taking a derivative of a table is extremely difficult...)
The analytical method is quite familiar and does not create problems. Perhaps there are some variations of this method that students encounter. I'm talking about parametric and implicit functions.) But such functions are in a special lesson.
Let's move on to less familiar ways of specifying a function.
Tabular method of specifying a function.
As the name suggests, this method is a simple sign. In this table, each x corresponds to ( is put in accordance) some meaning of the game. The first line contains the values of the argument. The second line contains the corresponding function values, for example:
Table 1.
| x | - 3 | - 1 | 0 | 2 | 3 | 4 |
| y | 5 | 2 | - 4 | - 1 | 6 | 5 |
Please pay attention! In this example, the game depends on X anyhow. I came up with this on purpose.) There is no pattern. It's okay, it happens. Means, exactly I have specified this specific function. Exactly I established a rule according to which an X turns into a Y.
You can make up another a plate containing a pattern. This sign will indicate other function, for example:
Table 2.
| x | - 3 | - 1 | 0 | 2 | 3 | 4 |
| y | - 6 | - 2 | 0 | 4 | 6 | 8 |
Did you catch the pattern? Here all the values of the game are obtained by multiplying x by two. Here is the first “tricky” question: can a function defined using Table 2 be considered a function y = 2x? Think for now, the answer will be below, in a graphical way. It's all very clear there.)
What's good tabular method of specifying a function? Yes, because you don’t need to count anything. Everything has already been calculated and written in the table.) But there is nothing more good. We don't know the value of the function for X's, which are not in the table. In this method, such x values are simply does not exist. By the way, this is a hint to a tricky question.) We cannot find out how the function behaves outside the table. We can't do anything. And the clarity of this method leaves much to be desired... The graphical method is good for clarity.
Graphical way to specify a function.
In this method, the function is represented by a graph. The argument (x) is plotted along the abscissa axis, and the function value (y) is plotted along the ordinate axis. According to the schedule, you can also choose any X and find the corresponding value at. The graph can be any, but... not just any one.) We work only with unambiguous functions. The definition of such a function clearly states: each X is put in accordance the only one at. One one game, not two, or three... For example, let's look at the circle graph:
A circle is like a circle... Why shouldn't it be the graph of a function? Let's find which game will correspond to the value of X, for example, 6? We move the cursor over the graph (or touch the drawing on the tablet), and... we see that this x corresponds two game meanings: y=2 and y=6.
Two and six! Therefore, such a graph will not be a graphical assignment of the function. On one x accounts for two game. This graph does not correspond to the definition of a function.
But if the unambiguity condition is met, the graph can be absolutely anything. For example:
This same crookedness is the law by which an X can be converted into a Y. Unambiguous. We wanted to know the meaning of the function for x = 4, For example. We need to find the four on the x-axis and see which game corresponds to this x. We move the mouse over the figure and see that the function value at For x=4 equals five. We don’t know what formula determines this transformation of an X into a Y. And it is not necessary. Everything is set by the schedule.
Now we can return to the “tricky” question about y=2x. Let's plot this function. Here he is:
Of course, when drawing this graph we did not take an infinite number of values X. We took several values and calculated y, made a sign - and everything is ready! The most literate people took only two values of X! And rightly so. For a straight line you don’t need more. Why the extra work?
But we knew for sure what x could be anyone. Integer, fractional, negative... Any. This is according to the formula y=2x it is seen. Therefore, we boldly connected the points on the graph with a solid line.
If the function is given to us by Table 2, then we will have to take the values of x only from the table. Because other X's (and Y's) are not given to us, and there is nowhere to take them. These values are not present in this function. The schedule will work out from points. We move the mouse over the figure and see the graph of the function given by Table 2. I didn’t write the x-y values on the axes, can you figure it out, cell by cell?)
Here is the answer to the “tricky” question. Function specified by Table 2 and function y=2x - different.
The graphical method is good for its clarity. You can immediately see how the function behaves, where it increases. where it decreases. From the graph you can immediately find out some important characteristics of the function. And in the topic with derivatives, tasks with graphs are all over the place!
In general, analytical and graphical methods of defining a function go hand in hand. Working with the formula helps to build a graph. And the graph often suggests solutions that you wouldn’t even notice in the formula... We will be friends with graphs.)
Almost any student knows the three ways to define a function that we just looked at. But to the question: “And the fourth!?” - freezes thoroughly.)
There is such a way.
Verbal description of the function.
Yes Yes! The function can be quite unambiguously specified in words. The great and mighty Russian language is capable of a lot!) Let's say the function y=2x can be specified with the following verbal description: Each real value of the argument x is associated with its double value. Like this! The rule is established, the function is specified.
Moreover, you can verbally specify a function that is extremely difficult, if not impossible, to define using a formula. For example: Each value of the natural argument x is associated with the sum of the digits that make up the value of x. For example, if x=3, That y=3. If x=257, That y=2+5+7=14. And so on. It is problematic to write this down in a formula. But the sign is easy to make. And build a schedule. By the way, the graph looks funny...) Try it.
The method of verbal description is quite exotic. But sometimes it does. I brought it here to give you confidence in unexpected and unusual situations. You just need to understand the meaning of the words "function specified..." Here it is, this meaning:
If there is a law of one-to-one correspondence between X And at- that means there is a function. What law, in what form it is expressed - a formula, a tablet, a graph, words, songs, dances - does not change the essence of the matter. This law allows you to determine the corresponding value of the Y from the value of X. All.
Now we will apply this deep knowledge to some non-standard tasks.) As promised at the beginning of the lesson.
Exercise 1:
The function y = f(x) is given by Table 1:
Table 1.
Find the value of the function p(4), if p(x)= f(x) - g(x)
If you can’t understand what’s what at all, read the previous lesson “What is a function?” It is written very clearly about such letters and brackets.) And if only the tabular form confuses you, then we’ll sort it out here.
From the previous lesson it is clear that if, p(x) = f(x) - g(x), That p(4) = f(4) - g(4). Letters f And g means the rules according to which each X is assigned its own game. For each letter ( f And g) - yours rule. Which is given by the corresponding table.
Function value f(4) determined from Table 1. This will be 5. Function value g(4) determined according to Table 2. This will be 8. The most difficult thing remains.)
p(4) = 5 - 8 = -3
This is the correct answer.
Solve the inequality f(x) > 2
That's it! It is necessary to solve the inequality, which (in the usual form) is brilliantly absent! All that remains is to either give up the task or turn on your head. We choose the second and discuss.)
What does it mean to solve inequality? This means finding all the values of x at which the condition given to us is satisfied f(x) > 2. Those. all function values ( at) must be greater than two. And on our chart we have every game... And there are more twos, and less... And let’s, for clarity, draw a border along this two! We move the cursor over the drawing and see this border.
Strictly speaking, this boundary is the graph of the function y=2, but that's not the point. The important thing is that now the graph shows very clearly where, at what X's, function values, i.e. y, more than two. They are more X > 3. At X > 3 our whole function passes higher borders y=2. That's the solution. But it’s too early to turn off your head!) I still need to write down the answer...
The graph shows that our function does not extend left and right to infinity. The points at the ends of the graph indicate this. The function ends there. Therefore, in our inequality, all x’s that go beyond the limits of the function have no meaning. For the function of these X's does not exist. And we, in fact, solve the inequality for the function...
The correct answer will be:
3 < X ≤ 6
Or, in another form:
X ∈ (3; 6]
Now everything is as it should be. Three is not included in the answer, because the original inequality is strict. And the six turns on, because and the function at six exists, and the inequality condition is satisfied. We have successfully solved an inequality that (in the usual form) does not exist...
This is how some knowledge and elementary logic saves you in non-standard cases.)
