Power function with even exponent. Power function, its properties and graphs. Power function with irrational exponent

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 range of values of power functions of the form, where n is a 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 a rational positive exponent. The domain of definition is all positive numbers and the number 0. The range of values of functions with such an 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

Are you familiar with the functions y=x, y=x 2 , y=x 3 , y=1/x etc. All these functions are special cases of the power function, i.e. the function y=xp, where p is a given real number.
The properties and graph of a power function significantly depend on the properties of a power with a real exponent, and in particular on the values for which x And p degree makes sense x p. Let us proceed to a similar consideration of various cases depending on
exponent p.

Index p=2n-even natural number.

y=x2n, Where n- a natural number, has the following

properties:

domain of definition - all real numbers, i.e. the set R;
set of values - non-negative numbers, i.e. y is greater than or equal to 0;
function y=x2n even, because x 2n=(- x) 2n
the function is decreasing on the interval x<0 and increasing on the interval x>0.

Graph of a function y=x2n has the same form as, for example, the graph of a function y=x 4.

2. Indicator p=2n-1- odd natural number
In this case, the power function y=x2n-1, where is a natural number, has the following properties:

domain of definition - set R;
set of values - set R;
function y=x2n-1 odd because (- x) 2n-1=x2n-1;
the function is increasing on the entire real axis.

Graph of a function y=x 2n-1 has the same form as, for example, the graph of the function y=x 3 .

3.Indicator p=-2n, Where n- natural number.

In this case, the power function y=x -2n =1/x 2n has the following properties:

domain of definition - set R, except x=0;
set of values - positive numbers y>0;
function y =1/x2n even, because 1/(-x)2n=1/x 2n;
the function is increasing on the interval x<0 и убывающей на промежутке x>0.

Graph of function y =1/x2n has the same form as, for example, the graph of the function y =1/x 2.

Lecture: Power function with natural exponent, its graph

We deal with functions all the time where the argument has some degree:
y = x 1, y = x 2, y = x 3, y = x -1, etc.

Graphs of power functions

So now we will look at several possible cases of a power function.

1) y = x 2 n .

This means that now we will consider functions in which the exponent is an even number.

Function Characteristic:

1. All real numbers are accepted as the range of values.

2. The function can accept all positive values and the number zero.

3. The function is even because it does not depend on the sign of the argument, but depends only on its modulus.

4. For a positive argument the function increases, and for a negative argument it decreases.

The graphs of these functions resemble a parabola. For example, below is a graph of the function y = x 4.

2) The function has an odd exponent: y = x 2 n +1.

1. The domain of a function is the entire set of real numbers.

2. Function value range - can take the form of any real number.

3. This function is odd.

4. Increases monotonically over the entire interval of consideration of the function.

5. The graph of all power functions with an odd exponent is identical to the function y = x 3.

3) The function has an even negative natural exponent: y = x -2 n.

We all know that a negative exponent allows us to omit the degree from the denominator and change the sign of the exponent, that is, we get the form y = 1/x 2 n.

1. The argument of this function can take any value except zero, since the variable is in the denominator.

2. Since the exponent is an even number, the function cannot take negative values. And since the argument cannot be equal to zero, then the value of the function equal to zero should also be excluded. This means that the function can only take positive values.

3. This function is even.

4. For a negative argument, the function increases monotonically, and for a positive argument, it decreases.

Type of graph of the function y = x -2: