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. (Trigonometric formulas).
19. Basic elementary functions, their properties and graphs. Application of functions in economics.
Basic elementary functions. Their properties and graphs
1. Linear function.
Linear function is called a function of the form , where x is a variable, a and b are real numbers.
Number A called the slope of the line, it is equal to the tangent of the angle of inclination of this line to the positive direction of the x-axis. The graph of a linear function is a straight line. It is defined by two points.
Properties of a Linear Function
1. Domain of definition - the set of all real numbers: D(y)=R
2. The set of values is the set of all real numbers: E(y)=R
3. The function takes a zero value when or.
4. The function increases (decreases) over the entire domain of definition.
5. A linear function is continuous over the entire domain of definition, differentiable and .
2. Quadratic function.
A function of the form, where x is a variable, coefficients a, b, c are real numbers, is called quadratic
1. Fractional linear function and its graph
A function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials, is called a fractional rational function.
You are probably already familiar with the concept of rational numbers. Likewise rational functions are functions that can be represented as the quotient of two polynomials.
If a fractional rational function is the quotient of two linear functions - polynomials of the first degree, i.e. function of the form
y = (ax + b) / (cx + d), then it is called fractional linear.
Note that in the function y = (ax + b) / (cx + d), c ≠ 0 (otherwise the function becomes linear y = ax/d + b/d) and that a/c ≠ b/d (otherwise the function is constant ). The linear fractional function is defined for all real numbers except x = -d/c. Graphs of fractional linear functions do not differ in shape from the graph y = 1/x you know. A curve that is a graph of the function y = 1/x is called hyperbole. With an unlimited increase in x in absolute value, the function y = 1/x decreases unlimited in absolute value and both branches of the graph approach the abscissa: the right one approaches from above, and the left one from below. The lines to which the branches of a hyperbola approach are called its asymptotes.
Example 1.
y = (2x + 1) / (x – 3).
Solution.
Let's select the whole part: (2x + 1) / (x – 3) = 2 + 7/(x – 3).
Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: shift by 3 unit segments to the right, stretching along the Oy axis 7 times and shifting by 2 unit segments upward.
Any fraction y = (ax + b) / (cx + d) can be written in a similar way, highlighting the “integer part”. Consequently, the graphs of all fractional linear functions are hyperbolas, shifted in various ways along the coordinate axes and stretched along the Oy axis.
To construct a graph of any arbitrary fractional-linear function, it is not at all necessary to transform the fraction defining this function. Since we know that the graph is a hyperbola, it will be enough to find the straight lines to which its branches approach - the asymptotes of the hyperbola x = -d/c and y = a/c.
Example 2.
Find the asymptotes of the graph of the function y = (3x + 5)/(2x + 2).
Solution.
The function is not defined, at x = -1. This means that the straight line x = -1 serves as a vertical asymptote. To find the horizontal asymptote, let’s find out what the values of the function y(x) approach when the argument x increases in absolute value.
To do this, divide the numerator and denominator of the fraction by x:
y = (3 + 5/x) / (2 + 2/x).
As x → ∞ the fraction will tend to 3/2. This means that the horizontal asymptote is the straight line y = 3/2.
Example 3.
Graph the function y = (2x + 1)/(x + 1).
Solution.
Let’s select the “whole part” of the fraction:
(2x + 1) / (x + 1) = (2x + 2 – 1) / (x + 1) = 2(x + 1) / (x + 1) – 1/(x + 1) =
2 – 1/(x + 1).
Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: a shift by 1 unit to the left, a symmetrical display with respect to Ox and a shift by 2 unit segments up along the Oy axis.
Domain D(y) = (-∞; -1)ᴗ(-1; +∞).
Range of values E(y) = (-∞; 2)ᴗ(2; +∞).
Intersection points with axes: c Oy: (0; 1); c Ox: (-1/2; 0). The function increases at each interval of the domain of definition.
Answer: Figure 1.
2. Fractional rational function
Consider a fractional rational function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials of degree higher than first.
Examples of such rational functions:
y = (x 3 – 5x + 6) / (x 7 – 6) or y = (x – 2) 2 (x + 1) / (x 2 + 3).
If the function y = P(x) / Q(x) represents the quotient of two polynomials of degree higher than the first, then its graph will, as a rule, be more complex, and it can sometimes be difficult to construct it accurately, with all the details. However, it is often enough to use techniques similar to those we have already introduced above.
Let the fraction be a proper fraction (n< m). Известно, что любую несократимую рациональную дробь можно представить, и притом единственным образом, в виде суммы конечного числа элементарных дробей, вид которых определяется разложением знаменателя дроби Q(x) в произведение действительных сомножителей:
P(x)/Q(x) = A 1 /(x – K 1) m1 + A 2 /(x – K 1) m1-1 + … + A m1 /(x – K 1) + …+
L 1 /(x – K s) ms + L 2 /(x – K s) ms-1 + … + L ms /(x – K s) + …+
+ (B 1 x + C 1) / (x 2 +p 1 x + q 1) m1 + … + (B m1 x + C m1) / (x 2 +p 1 x + q 1) + …+
+ (M 1 x + N 1) / (x 2 +p t x + q t) m1 + … + (M m1 x + N m1) / (x 2 +p t x + q t).
Obviously, the graph of a fractional rational function can be obtained as the sum of graphs of elementary fractions.
Plotting graphs of fractional rational functions
Let's consider several ways to construct graphs of a fractional rational function.
Example 4.
Draw a graph of the function y = 1/x 2 .
Solution.
We use the graph of the function y = x 2 to construct a graph of y = 1/x 2 and use the technique of “dividing” the graphs.
Domain D(y) = (-∞; 0)ᴗ(0; +∞).
Range of values E(y) = (0; +∞).
There are no points of intersection with the axes. The function is even. Increases for all x from the interval (-∞; 0), decreases for x from 0 to +∞.
Answer: Figure 2.
Example 5.
Graph the function y = (x 2 – 4x + 3) / (9 – 3x).
Solution.
Domain D(y) = (-∞; 3)ᴗ(3; +∞).
y = (x 2 – 4x + 3) / (9 – 3x) = (x – 3)(x – 1) / (-3(x – 3)) = -(x – 1)/3 = -x/ 3 + 1/3.
Here we used the technique of factorization, reduction and reduction to a linear function.
Answer: Figure 3.
Example 6.
Graph the function y = (x 2 – 1)/(x 2 + 1).
Solution.
The domain of definition is D(y) = R. Since the function is even, the graph is symmetrical about the ordinate. Before building a graph, let’s transform the expression again, highlighting the whole part:
y = (x 2 – 1)/(x 2 + 1) = 1 – 2/(x 2 + 1).
Note that isolating the integer part in the formula of a fractional rational function is one of the main ones when constructing graphs.
If x → ±∞, then y → 1, i.e. the straight line y = 1 is a horizontal asymptote.
Answer: Figure 4.
Example 7.
Let's consider the function y = x/(x 2 + 1) and try to accurately find its largest value, i.e. the highest point on the right half of the graph. To accurately construct this graph, today's knowledge is not enough. Obviously, our curve cannot “rise” very high, because the denominator quickly begins to “overtake” the numerator. Let's see if the value of the function can be equal to 1. To do this, we need to solve the equation x 2 + 1 = x, x 2 – x + 1 = 0. This equation has no real roots. This means our assumption is incorrect. To find the largest value of the function, you need to find out at what largest A the equation A = x/(x 2 + 1) will have a solution. Let's replace the original equation with a quadratic one: Ax 2 – x + A = 0. This equation has a solution when 1 – 4A 2 ≥ 0. From here we find the largest value A = 1/2. 
Answer: Figure 5, max y(x) = ½.
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Definition: A numerical function is a correspondence that associates each number x from some given set with a single number y.
Designation:
where x is the independent variable (argument), y is the dependent variable (function). The set of values of x is called the domain of the function (denoted D(f)). The set of values of y is called the range of values of the function (denoted E(f)). The graph of a function is the set of points in the plane with coordinates (x, f(x))
Methods for specifying a function.
- analytical method (using a mathematical formula);
- tabular method (using a table);
- descriptive method (using verbal description);
- graphical method (using a graph).
Basic properties of the function.
1. Even and odd
A function is called even if
– the domain of definition of the function is symmetrical about zero
f(-x) = f(x)
The graph of an even function is symmetrical about the axis 0y
A function is called odd if
– the domain of definition of the function is symmetrical about zero
– for any x from the domain of definition f(-x) = –f(x)
The graph of an odd function is symmetrical about the origin.
2. Frequency
A function f(x) is called periodic with period if for any x from the domain of definition f(x) = f(x+T) = f(x-T) .
The graph of a periodic function consists of unlimitedly repeating identical fragments.
3. Monotony (increasing, decreasing)
The function f(x) is increasing on the set P if for any x 1 and x 2 from this set such that x 1
The function f(x) decreases on the set P if for any x 1 and x 2 from this set, such that x 1 f(x 2) .
4. Extremes
The point X max is called the maximum point of the function f(x) if for all x from some neighborhood of X max the inequality f(x) f(X max) is satisfied.
The value Y max =f(X max) is called the maximum of this function.
X max – maximum point
At max - maximum
A point X min is called a minimum point of the function f(x) if for all x from some neighborhood of X min, the inequality f(x) f(X min) is satisfied.
The value Y min =f(X min) is called the minimum of this function.
X min – minimum point
Y min – minimum
X min , X max – extremum points
Y min , Y max – extrema.
5. Zeros of the function
The zero of a function y = f(x) is the value of the argument x at which the function becomes zero: f(x) = 0.
X 1, X 2, X 3 – zeros of the function y = f(x).
Tasks and tests on the topic "Basic properties of a function"
- Function Properties - Numerical functions 9th grade
Lessons: 2 Assignments: 11 Tests: 1
- Properties of logarithms
Lessons: 2 Assignments: 14 Tests: 1
- Square root function, its properties and graph - Square root function. Properties of square root grade 8
Lessons: 1 Assignments: 9 Tests: 1
- Power functions, their properties and graphs - Degrees and roots. Power functions grade 11
Lessons: 4 Assignments: 14 Tests: 1
- Exponential function, its properties and graph - Exponential and logarithmic functions grade 11
Lessons: 1 Assignments: 15 Tests: 1
Having studied this topic, you should be able to find the domain of definition of various functions, determine the monotonicity intervals of a function using graphs, and examine functions for evenness and oddness. Let's consider solving similar problems using the following examples.
Examples.
1. Find the domain of definition of the function.
Solution: the domain of definition of the function is found from the condition
therefore, the function f(x) is even.
Answer: even
D(f) = [-1; 1] – symmetrical about zero.
| 2) |
hence the function is neither even nor odd.
Answer: neither even nor uneven.
