Y is equal to the root of X. Power function and roots - definition, properties and formulas

Lesson and presentation on the topic: "Power functions. Cubic root. Properties of the cubic root"

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Definition of a power function - cube root

Let's check the equality: $\sqrt((-x))$=-$\sqrt(x)$.
Let $\sqrt((-x))=a$ and $\sqrt(x)=b$. Let's raise both expressions to the third power. $–x=a^3$ and $x=b^3$. Then $a^3=-b^3$ or $a=-b$. In the notation of roots we obtain the desired identity.

Properties of cube roots

Let's prove the second property. $(\sqrt(\frac(a)(b)))^3=\frac(\sqrt(a)^3)(\sqrt(b)^3)=\frac(a)(b)$.
We found that the number $\sqrt(\frac(a)(b))$ cubed is equal to $\frac(a)(b)$ and then equals $\sqrt(\frac(a)(b))$, which and needed to be proven.

Guys, let's build a graph of our function.
1) Domain of definition is the set of real numbers.
2) The function is odd, since $\sqrt((-x))$=-$\sqrt(x)$. Next, consider our function for $x≥0$, then display the graph relative to the origin.
3) The function increases when $x≥0$. For our function, the larger argument value corresponds to higher value functions, which means increasing.
4) The function is not limited from above. In fact, from any large number the third root can be calculated, and we can move upward indefinitely, finding ever larger values ​​of the argument.
5) For $x≥0$ smallest value equals 0. This property is obvious.
Let's build a graph of the function by points at x≥0.

Let's construct our graph of the function over the entire domain of definition. Remember that our function is odd.

Function properties:
1) D(y)=(-∞;+∞).
2) Odd function.
3) Increases by (-∞;+∞).
4) Unlimited.
5) There is no minimum or maximum value.

7) E(y)= (-∞;+∞).
8) Convex downward by (-∞;0), convex upward by (0;+∞).

Examples of solving power functions

2. Construct a graph of the function. $y=\sqrt((x-2))-3$.
Solution. Our graph is obtained from the graph of the function $y=\sqrt(x)$, parallel transfer two units to the right and three units down.

3. Graph the function and read it. $\begin(cases)y=\sqrt(x), x≥-1\\y=-x-2, x≤-1 \end(cases)$.
Solution. Let's construct two graphs of functions on the same coordinate plane, taking into account our conditions. For $x≥-1$ we build a graph of the cubic root, for $x≤-1$ we build a graph of a linear function.
1) D(y)=(-∞;+∞).
2) The function is neither even nor odd.
3) Decreases by (-∞;-1), increases by (-1;+∞).
4) Unlimited from above, limited from below.
5) There is no greatest value. The smallest value is minus one.
6) The function is continuous on the entire number line.
7) E(y)= (-1;+∞).

Problems to solve independently

Nth degree of real number, noted that the root of any degree (second, third, fourth, etc.) can be extracted from any non-negative number, and the root of any odd degree can be extracted from a negative number. But then you should think about a function of the form , about its graph, about its properties. This is what we will do in this paragraph. First let's talk about the function in case of non-negative values argument.

Let's start with the case you know, when n = 2, i.e. from the function In Fig. 166 shows the graph of the function and the graph of the function y = x 2, x>0. Both graphs represent the same curve - a branch of a parabola, only located differently on the coordinate plane. Let us clarify: these graphs are symmetrical relative to the straight line y = x, since they consist of points that are symmetrical to each other relative to the specified straight line. Look: on the considered branch of the parabola y = x 2 there are points (0; 0), (1; 1), (2; 4), (3; 9), (4; 16), and on the function graph there are points (0; 0), (1; 1), (4; 2), (9; 3), (16; 4).

Points (2; 4) and (4; 2), (3; 9) and (9; 3), (4; 16) and (16; 4) are symmetrical about the line y = x, (and points (0; 0 ) and (1; 1) lie on this line). And in general, for any point (a; a 2) on function graph y = x 2 is a point (a 2 ; a) symmetrical to it with respect to the straight line y = x on the graph of the function and vice versa. The following theorem is true.

Proof. For definiteness, we assume that a and b are positive numbers. Consider the triangles OAM and OVR (Fig. 167). They are equal, which means OP = OM and . But then since the straight line y = x is the bisector of the angle AOB. So, the triangle ROM is isosceles, OH is its bisector, and therefore the axis of symmetry. Points M and P are symmetrical with respect to straight line OH, which is what needed to be proven.
So, the graph of the function can be obtained from the graph of the function y = x 2, x>0 using a symmetry transformation about the straight line y = x. Similarly, the graph of a function can be obtained from the graph of the function y = x 3, x> 0 using a symmetry transformation about the straight line y = x; the graph of a function can be obtained from the graph of a function using a symmetry transformation about the straight line y = x, etc. Let us recall that the graph of a function resembles in appearance the branch of a parabola. The larger n, the steeper this branch rushes upward in the interval and the closer it approaches the x axis in the vicinity of the point x = 0 (Fig. 168).

Let us formulate a general conclusion: the graph of the function is symmetrical to the graph of the function relative to the straight line y = x (Fig. 169).

Function properties

1)
2) the function is neither even nor odd;
3) increases by
4) not limited from above, limited from below;
5) does not have the greatest significance;
6) continuous;
7)

Pay attention to one curious circumstance. Let's consider two functions, the graphs of which are shown in Fig. 169: We have just listed seven properties for the first function, but the second function has absolutely the same properties. Verbal “portraits” of two various functions are the same. But, let’s clarify, they are still the same.

Mathematicians could not bear such an injustice when different functions with different graphs are verbally described in the same way, and introduced the concepts of upward convexity and downward convexity. The graph of the function is convex upward, while the graph of the function y = x n is convex downward.

It is usually said that a continuous function is convex downward if, by connecting any two points of its graph with a straight line segment, it is discovered that the corresponding part of the graph lies below the drawn segment (Fig. 170); a continuous function is convex upward if, by connecting any two points of its graph with a straight line segment, it is discovered that the corresponding part of the graph lies above the drawn segment (Fig. 171).

We will further include the convexity property in the procedure for reading a graph. Let us note it" (continuing the numbering of the properties described earlier) for the function under consideration:

8) the function is convex upward on the ray
In the previous chapter, we became acquainted with another property of a function - differentiability; we saw that the function y = x n is differentiable at any point, its derivative is equal to nx n-1. Geometrically, this means that at any point on the graph of the function y = x n a tangent can be drawn to it. The graph of a function also has the same property: at any point it is possible to draw a tangent to the graph. Thus, we can note one more property of the function
9) the function is differentiable at any point x > 0.
Please note: we are not talking about the differentiability of the function at the point x = 0 - at this point the tangent to the graph of the function coincides with the y-axis, i.e. perpendicular to the x-axis.
Example 1. Graph a function
Solution. 1) Let's move on to auxiliary system coordinates with the origin at point (-1; -4) - dotted lines x = -1 and y = -4 in Fig. 172.
2) “Bind” the function to new system coordinates This will be the required schedule.
Example 2. Solve the equation

Solution. First way. 1) Let us introduce two functions
2) Let's plot the function

3) Let's build a graph of the linear function y=2-x (see Fig. 173).

4) The constructed graphs intersect at one point A, and from the graph we can make the assumption that the coordinates of point A are as follows: (1; 1). The check shows that in fact the point (1; 1) belongs to both the graph of the function and the graph of the function y=2-x. This means that our equation has one root: x = 1 - the abscissa of point A.

Second way.
The geometric model presented in Fig. 173, is clearly illustrated by the following statement, which sometimes allows you to solve the equation very elegantly (and which we already used in § 35 when solving Example 2):

If the function y=f(x) increases, and the function y=g(x) decreases, and if the equation f(x)=g(x) has a root, then there is only one.

Here's how, based on this statement, we can solve the given equation:

1) note that for x = 1 the equality is satisfied, which means x = 1 is the root of the equation (we guessed this root);
2) the function y=2-x decreases, and the function increases; This means that the given equation has only one root, and this root is the value x = 1 found above.

Answer: x = 1.

So far we have talked about the function only for non-negative argument values. But if n is an odd number, the expression also makes sense for x<0. Значит, есть смысл поговорить о функции в случае нечетного п для любых значений х.

As a matter of fact, only one property will be added to those listed:

if n is an odd number (n = 3.5, 7,...), then it is an odd function.

In fact, let such transformations be true for an odd exponent n. So, f(-x) = -f(x), and this means the function is odd.

What does the graph of a function look like in the case of an odd exponent n? When as shown in Fig. 169, is a branch of the desired graph. By adding to it a branch that is symmetrical to it relative to the origin of coordinates (which, recall, is typical for any odd function), we obtain a graph of the function (Fig. 174). Note that the y-axis is tangent to the graph at x = 0.
So let's repeat it again:
if n is an even number, then the graph of the function has the form shown in Fig. 169;
if n is an odd number, then the graph of the function has the form shown in Fig. 174.

Example 3. Construct and read a graph of the function y = f(x), where
Solution. First, let's build a graph of the function and highlight part of it on the ray (Fig. 175).
Then we will construct a graph of the function and select its part on the open beam (Fig. 176). Finally, we will depict both “pieces” in the same coordinate system - this will be the graph of the function y = f(x) (Fig. 177).
Let us list (based on the plotted graph) the properties of the function y = f(x):

1)
2) neither even nor odd;
3) decreases on the ray, increases on the ray
4) not limited from below, limited from above;
5) there is no minimum value, a (achieved at point x = 1);
6) continuous;
7)
8) convex downwards at , convex upwards on the segment , convex downwards at
9) the function is differentiable everywhere except for the points x = 0 and x = 1.
10) the graph of the function has a horizontal asymptote, which means, recall that

Example 4. Find the domain of a function:

Solution, a) Under the sign of the root of an even degree there must be a non-negative number, which means the problem comes down to solving the inequality
b) Any number can be under the sign of an odd root, which means that here no restrictions are imposed on x, i.e. D(f) = R.
c) The expression makes sense provided that a expression means that two inequalities must be satisfied simultaneously: those. the problem comes down to solving the system of inequalities:

Solving Inequalities
Let's solve the inequality Let's factorize the left side of the inequality: The left side of the inequality turns to 0 at points -4 and 4. Let's mark these points on the number line (Fig. 178). The number line is divided by the indicated points into three intervals, and at each interval the expression p(x) = (4-x)(4 + x) retains a constant sign (the signs are indicated in Fig. 178). The interval over which the inequality p(x)>0 holds is shaded in Fig. 178. According to the conditions of the problem, we are also interested in those points x at which the equality p(x) = 0 holds. There are two such points: x = -4, x = 4 - they are marked in Fig. 178 dark circles. Thus, in Fig. 178 presents a geometric model for solving the second inequality of the system.

Let us mark the found solutions to the first and second inequalities of the system on the same coordinate line, using the upper hatch for the first and the lower hatch for the second (Fig. 179). The solution to the system of inequalities will be the intersection of the solutions to the system’s inequalities, i.e. the interval where both hatchings coincide. Such a gap is the segment [-1, 4].

Answer. D(f) = [-1.4].

A.G. Mordkovich Algebra 10th grade

Calendar-thematic planning in mathematics, video in mathematics online, Mathematics at school

Consider the function y=√x. The graph of this function is shown in the figure below.

Graph of the function y=√x

As you can see, the graph resembles a rotated parabola, or rather one of its branches. We get a branch of the parabola x=y^2. It can be seen from the figure that the graph touches the Oy axis only once, at the point with coordinates (0;0).
Now it is worth noting the main properties of this function.

Properties of the function y=√x

1. The domain of definition of a function is a ray)