Presentation for an algebra lesson (grade 10) on the topic: Numerical functions. Definition and methods of assignment. “Definition of a numerical function and how to define it” - Lesson
The figure shows a correspondence graph between sets X= {A;b;With;d;e},Y = (1; 2; 3; 4; 5). This correspondence is such that not every element of the set X there is a corresponding element of the set Y, but if there is, then he is the only one.
A= {A;b;With) – the set of those elements for which there is a corresponding element in the set Y. Note that each element of the set A corresponds to the only element of the set Y.
Definition. Correspondence between sets X And Y, where each element of the set X corresponds to at most one element of the set Y, is called a functional correspondence or function.
Functions are indicated by letters Latin alphabet f,g,h etc. and write: at=f(X).
X– independent variable or argument, all values that the independent variable takes – the domain of definition of the function.
Let the function be given f with domain AX, Where X– function departure set f. We denote the arrival set Y.
Element at Y, corresponding to the element XA, is called the value of the function f and write at=f(X).
Definition. Plenty of everyone at Y, which are the function values f, is called the set of function values f.
If a function is specified by a formula and its domain of definition is not specified, then the domain of definition of the function is considered to consist of all values of the argument for which the formula makes sense.
Example. Let the function be given f(X)
=
. Function definition domain f(X) is a set R
\ {2}.
Methods for specifying functions
Analytical assignment of a function - assignment of a function using a formula at=f(X), Where f(X) – some expression in a variable X.
Tabular specification of a function - a table is provided indicating the value of the function for the argument values available in the table. This method is often used in practice when the dependence of one quantity on another is found experimentally; turns out to be convenient, because allows you to find the value of a function for the argument values available in the table without calculations.
Graphical function specification. 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 Properties
Even and odd functions
Definition. Function at=f(X) is said to be even if for any element X f(–X) = f(X).
Definition. Function at=f(X) is called odd if for any element X from the domain of definition of the function the equality f(–X) = – f(X).
From the definitions it follows that the domain of definition X both even and odd functions must have the following property: if XX, That - XX.
The graph of an even function is symmetrical about the ordinate axis, and the graph of an odd function is symmetrical about the origin.
Increasing and decreasing functions
Definition. Function at=f(X) is called increasing on the interval X, if X 1 ,X 2 X, such that X 1 <X 2, the inequality holds f(X 1) < f(X 2).
Definition. Function at=f(X) is called decreasing on the interval X, if X 1 ,X 2 X, such that X 1 <X 2, the inequality holds f(X 1) > f(X 2).
Definition. The function is called monotonic on a certain interval A, if it increases or decreases over this interval.
Numeric function is a function whose domain of definition (arguments) and range of values of the function are numerical sets. , where , are numerical sets.
An example of a numerical function is the dependence of your growth (function value) on time (argument) (Fig. 1).
Rice. 1. Growth function graph
The function that assigns each person his shoe size is not numeric because its arguments are not numbers.
Like any other objects, functions are usually classified to make them more convenient to study. You are familiar with different types of functions: linear, quadratic, logarithmic, etc. Let's look at the simplest functions - linear ones.
Equation of a linear function: , and are some numbers. The graph is straight (Fig. 2).
Rice. 2. Example of a graph of a linear function
Why can a linear function be called simple? Since its graph is a straight line. Any non-vertical straight line on the coordinate plane defines a linear function and vice versa. In geometry, a straight line is one of the simplest objects.
In addition, we often encounter and use linear functions in life. For example, when we say that a car is moving at a speed of km/h. This means that in the first hour he will travel km, in the second - km, etc. That is, the same changes in the argument (time) lead to the same change in the function (the distance the car has traveled).
Let us describe the movement of the car: let the initial position be , and in hours at a constant speed it will cover a distance . Then the position of the car at a given time will be determined as follows: , where is the argument of the function.
This equation describes a linear function. Let's take two moments in time and:
We see that the change in the value of a function is proportional to the change in the value of its argument.
The linear function is also important because it can be used to locally approximate (describe) other functions. For example, if we take a small section of the graph (Fig. 3) (Fig. 4), we will see that it is close to a straight line.
Rice. 3. Graph of a function
Rice. 4. Part of the graph in Fig. 3.
Having done this for the entire function, we obtained a piecewise linear function (Fig. 5). Now we can describe its behavior on each linear section.
Rice. 5. Piecewise linear function
A simple example of approximating a curved line using short straight segments is studied in computer science at school: a turtle draws a circle in this way in the LOGO program. It is clear that it is impossible to draw an ideal circle on the screen: the screen has a minimum cell (pixel). We call it a point, but it still has some width and length. And it is clear that it is impossible to draw a smooth circle - in fact, the result will be a very, very accurate, but still approximation.
If we look at a photo on the screen, the lines seem to be smooth. But if you start to increase it, then sooner or later squares (pixels) become visible (Fig. 6).
Rice. 6. Enlarging the photo on the screen
The same can be seen in the circle drawn by the turtle. Upon magnification, it will become noticeable that what is actually drawn is not a circle, but a regular n-gon with a sufficiently large value (Fig. 7).
Rice. 7. Enlarged image of a circle
In life we often use this method. For example, when watching a bird fly, we unconsciously calculate its speed and assume that it will fly further in a straight line at the same speed (Fig. 8). In fact, our prediction may differ from reality, but over a short period of time it will be quite accurate.
Rice. 8. Illustration of miscalculation of the bird's position
We are not the only ones who perform this type of analysis. Many animals also know how to solve such problems: for example, when a frog catches a mosquito, it must be able to predict the point at which it will be in order to have time to throw out its tongue.
For more accurate measurements, we use more precise instruments. For functions, a more accurate tool (compared to a linear function) is the quadratic function. We can say that this is the next most difficult function.
Equation of a quadratic function: , where , and are some numbers.
The graph of a quadratic function is a parabola (Fig. 9).
Rice. 9. Example of a graph of a quadratic function
Using the quadratic function, we can more accurately approximate functions unknown to us, and therefore make more accurate predictions.
Another frequently encountered problem related to numerical functions: we know the values of a function at certain points, but we need to understand how the function behaves between these points. For example, we have some experimental data (Fig. 10).
Rice. 10. Experimental results
To understand how the air temperature behaved between the marked points, we need to somehow assume how the function behaves, since we cannot make an infinite number of measurements. You can approximate linearly (Fig. 11, graph A) or quadratically (Fig. 11, graph B).
Rice. 11. Linear and quadratic approximation
Such processes are called interpolation.
The task seems difficult: it may seem like fortune telling with coffee grounds. Indeed, we do not know how the function will behave between two marked points. For example, its graph may look like this (Fig. 12).
Rice. 12. “Unexpected” behavior of the function graph
In fact, we reconstruct the graph of the function point by point using some model: we assume that the function is sufficiently smooth if there were no sharp jumps in the model (for example, during an experiment). Then with a high degree of probability we can say that the graph of the function looks as shown in Fig. 11.
Quadratic and linear functions are united by the fact that they are specified by a polynomial (there are other such functions):
In addition to such functions, there are others; they describe various processes of physics and biology and are also studied. You can set them, describe their properties, build their graphs, and then work with them. Such functions include, for example, exponential, logarithmic, and trigonometric functions. We will talk about them in the next lessons.
Numeric function
This correspondence between a number set is called X and many R real numbers, in which each number from the set X matches a single number from a set R. Many X called domain of the function
. Functions are indicated by letters f, g, h etc. If f- a function defined on a set X, then real number y, corresponding to the number X there are many of them X, often denoted f(x) and write
y = f(x). Variable X this is called an argument. Set of numbers of the form f(x) called function range
The function is specified using a formula. For example , y = 2X - 2. If, when specifying a function using a formula, its domain of definition is not indicated, then it is assumed that the domain of definition of the function is the domain of definition of the expression f(x).
For example. If a function is given by the formula, then its domain of definition is the set of real numbers, excluding the number 2 (if x = 2, then the denominator of this fraction becomes zero).
Numerical functions can be represented visually using a graph on a coordinate plane. A graph is a set of points on the coordinate plane that have an abscissa X and ordinate f(x) for everyone X from many X. So, the graph of the function y = x + 2, defined on the set R, is a straight line (Fig. 1), and the graph of a function defined on the same set is a parabola (Fig. 2).
To construct a graph, you can use a table of corresponding values X And at:
1) for function y = x + 2
2) for function
Not every set of points on a coordinate plane represents a graph of some function. Since for each value of the argument from the domain of definition the function must have only one value, then any straight line parallel to the ordinate axis either does not intersect the graph of the function at all, or intersects it only at one point. If this condition is not met, then the set of points on the coordinate plane does not define the graph of the function.
For example, the curve in Fig. 3.
Functions can be specified using either a graph or a table. For example, the table below describes the dependence of air temperature on the time of day. This dependence is a function, since each time value t corresponds to a single air temperature value p.
t ( in hours) | ||||||||||
p(in degrees) |
What is a function? Definition. Correspondences in which each element of one set is associated with a single element of another set are called functions. They write: y = f(x), x Є X. The variable x is called an independent variable or argument. The set of all permissible values of the independent variable is the domain of the function and is denoted D(y). The variable y is the dependent variable. The set of all values of the dependent variable is the range of values of the function and is denoted E(y).
Methods for specifying a function There are 4 ways to specify a function. 1. Tabular method. It is convenient because it allows you to find the function values of the argument values available in the table without calculations. Х2345 У Analytical method. The function is specified by one or more formulas. This method is indispensable for studying a function and establishing its properties. Y=2 x+5, y= x² -5 x+1, y= |x+5|. 3. Graphic method. The function is specified by its geometric model on the coordinate plane. 4. Descriptive method. It is convenient to use when the task is difficult in other ways.
§3 Properties of the function Monotonicity: Increasing; decreasing function zeros (argument values in which the Function value is equal to zero) continuity periodicity even oddness Extrema: maximum point, minimum point convexity Maximum and minimum values of the function Intervals of constant sign (intervals in which the function takes only positive or only negative values)
A. A function of the form y=k/x, where k 0, is called inverse proportionality. The graph of inverse proportionality (hyperbola) is obtained from the graph of the function y = 1/x using stretching (and with k 
Function y = |x| y=|x |= x if x 0 -x if x 
0. O. The graph of a fractional-linear function is a hyperbola obtained from the graph of inverse proportionality using a shift." title=" Fractional-linear function O. A function of the form is called fractional-linear, where c>0. O. Graph fractional linear function - a hyperbola obtained from the inverse proportionality graph using a shift." class="link_thumb"> 11 !} Fractional-linear function O. A function of the form is called fractional-linear, where c>0. O. The graph of a fractional linear function is a hyperbola obtained from the graph of inverse proportionality using a shift. 0. O. Graph of a fractional-linear function - a hyperbola obtained from the graph of inverse proportionality using a shift."> 0. O. Graph of a fractional-linear function - a hyperbola obtained from the graph of inverse proportionality using a shift."> 0. O. The graph of a fractional-linear function is a hyperbola obtained from the graph of inverse proportionality using a shift." title="Fractional-linear function O. A function of the form is called fractional-linear, where c>0. O. Graph of a fractional-linear function - a hyperbola obtained from an inverse proportionality graph using a shift."> title="Fractional-linear function O. A function of the form is called fractional-linear, where c>0. O. The graph of a fractional linear function is a hyperbola obtained from the graph of inverse proportionality using a shift."> !}
Finding the Domain of a Function
Set of values of the function 1.у= 2sin²x-cos2x Solution: 2sin²x-cos2x=2sin²x-(1-2sin²x)=4sin²x-1 0 Sin²x 1, -1 4sin²x-1 3 Answer: -1 y 3 2. y = |cosx| Solution: -1 cosx 1, 0 |cosx| 1, |cosx| 1 1 Answer: -1 y 1 3. The function is given by a graph. Provide multiple values for this function. E(f)=(-2;2] E(f)= [-3;1] E(f)= (-;4]
To use presentation previews, create an account for yourself ( account) Google and log in: https://accounts.google.com
Slide captions:
Numerical functions. Definition and methods of assignment.
Let us recall If a numerical set and a rule are given that allows each element from the set to be assigned a certain number, then they say that a function with a domain of definition is given: – the domain of definition of the function; – independent variable or argument; – dependent variable; the set of all values is called the range of values of a function and is denoted by.
If a function is given, and all points of the form, where, a, are marked on the coordinate plane, then the set of these points is called the graph of the function, .
Graphs of some functions are straight
parabola
hyperbola
Knowing the graph of a function, you can construct a graph of the function using geometric transformations. To do this, you need to make a parallel transfer of the function graph to a vector, that is, to the right, if, and to the left, if, to up, if, and down, if.
Example -4 0 1 2 3 4
Set a function – specify a rule that allows you to calculate the corresponding value based on an arbitrarily selected value. Most often this rule is associated with a formula (for example). This method of specifying a function is called analytical.
Example Let be some line on the coordinate plane
Thus, a function is defined on the segment. This method of specifying a function is called graphical. Note that if the function was specified analytically and we were able to construct its graph, then we have actually made the transition from the analytical method of specifying the function to the graphical one.
The tabular method of specifying a function is using a table that indicates the values of the function for a finite set of argument values. For example: 5 7 8 9 10 12 5 7 4 6 5 7 8 9 10 12 5 7 4 6
The verbal method of specifying a function is a method in which the rule for specifying a function is described in words.
