What do the properties of a power function depend on? Power function

In this lesson, we will continue the study of power functions with rational indicator, consider functions with a negative rational exponent.

1. Basic concepts and definitions

Recall the properties and graphs of power functions with a negative integer exponent.

For even n, :

Function example:

All graphs of such functions pass through two fixed points: (1;1), (-1;1). A feature of functions of this type is their parity, the graphs are symmetrical with respect to the op-y axis.

Rice. 1. Graph of a function

For odd n, :

Function example:

All graphs of such functions pass through two fixed points: (1;1), (-1;-1). A feature of functions of this type is their oddness, the graphs are symmetrical with respect to the origin.

Rice. 2. Function Graph

2. Function with a negative rational exponent, graphs, properties

Let us recall the main definition.

The degree of a non-negative number a with a rational positive exponent is called a number.

The degree of a positive number a with a rational negative exponent is called a number.

For the following equality holds:

For example: ; - the expression does not exist by definition of a degree with a negative rational exponent; exists, since the exponent is an integer,

Let us turn to the consideration of power functions with a rational negative exponent.

For example:

To plot this function, you can make a table. We will do otherwise: first, we will build and study the graph of the denominator - we know it (Figure 3).

Rice. 3. Graph of a function

The graph of the denominator function passes through a fixed point (1;1). When constructing a graph of the original function, this point remains, when the root also tends to zero, the function tends to infinity. And, conversely, as x tends to infinity, the function tends to zero (Figure 4).

Rice. 4. Function Graph

Consider one more function from the family of functions under study.

It is important that by definition

Consider the graph of the function in the denominator: , we know the graph of this function, it increases in its domain of definition and passes through the point (1; 1) (Figure 5).

Rice. 5. Function Graph

When constructing a graph of the original function, the point (1; 1) remains, when the root also tends to zero, the function tends to infinity. And, conversely, as x tends to infinity, the function tends to zero (Figure 6).

Rice. 6. Function Graph

The considered examples help to understand how the graph goes and what are the properties of the function under study - a function with a negative rational exponent.

Graphs of functions of this family pass through the point (1;1), the function decreases over the entire domain of definition.

Function scope:

The function is not bounded from above, but bounded from below. The function has neither a maximum nor the smallest value.

The function is continuous, it takes all positive values ​​from zero to plus infinity.

Convex Down Function (Figure 15.7)

Points A and B are taken on the curve, a segment is drawn through them, the entire curve is below the segment, this condition is satisfied for arbitrary two points on the curve, therefore the function is convex downward. Rice. 7.

Rice. 7. Convexity of a function

3. Solution of typical problems

It is important to understand that the functions of this family are bounded from below by zero, but they do not have the smallest value.

Example 1 - find the maximum and minimum of a function on the interval \[(\mathop(lim)_(x\to +\infty ) x^(2n)\ )=+\infty \]

Graph (Fig. 2).

Figure 2. Graph of the function $f\left(x\right)=x^(2n)$

Properties of a power function with natural odd exponent

The domain of definition is all real numbers.

$f\left(-x\right)=((-x))^(2n-1)=(-x)^(2n)=-f(x)$ is an odd function.

$f(x)$ is continuous on the entire domain of definition.

The range is all real numbers.

$f"\left(x\right)=\left(x^(2n-1)\right)"=(2n-1)\cdot x^(2(n-1))\ge 0$

The function increases over the entire domain of definition.

$f\left(x\right)0$, for $x\in (0,+\infty)$.

$f(""\left(x\right))=(\left(\left(2n-1\right)\cdot x^(2\left(n-1\right))\right))"=2 \left(2n-1\right)(n-1)\cdot x^(2n-3)$

\ \

The function is concave for $x\in (-\infty ,0)$ and convex for $x\in (0,+\infty)$.

Graph (Fig. 3).

Figure 3. Graph of the function $f\left(x\right)=x^(2n-1)$

Power function with integer exponent

To begin with, we introduce the concept of a degree with an integer exponent.

Definition 3

Degree real number$a$ with integer index $n$ is determined by the formula:

Figure 4

Consider now a power function with an integer exponent, its properties and graph.

Definition 4

$f\left(x\right)=x^n$ ($n\in Z)$ is called a power function with integer exponent.

If the degree is greater than zero, then we come to the case of a power function with a natural exponent. We have already considered it above. For $n=0$ we get a linear function $y=1$. We leave its consideration to the reader. It remains to consider the properties of a power function with a negative integer exponent

Properties of a power function with a negative integer exponent

The scope is $\left(-\infty ,0\right)(0,+\infty)$.

If the exponent is even, then the function is even; if it is odd, then the function is odd.

$f(x)$ is continuous on the entire domain of definition.

Range of value:

If the exponent is even, then $(0,+\infty)$, if odd, then $\left(-\infty ,0\right)(0,+\infty)$.

If the exponent is odd, the function decreases as $x\in \left(-\infty ,0\right)(0,+\infty)$. For an even exponent, the function decreases as $x\in (0,+\infty)$. and increases as $x\in \left(-\infty ,0\right)$.

$f(x)\ge 0$ over the entire domain

Lesson and presentation on the topic: "Power functions. Properties. Graphs"

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Power functions, domain of definition.

Guys, in the last lesson we learned how to work with numbers with a rational exponent. In this lesson, we will consider power functions and restrict ourselves to the case when the exponent is rational.
We will consider functions of the form: $y=x^(\frac(m)(n))$.
Let us first consider functions whose exponent is $\frac(m)(n)>1$.
Let us be given a specific function $y=x^2*5$.
According to the definition we gave in the last lesson: if $x≥0$, then the domain of our function is the ray $(x)$. Let's schematically depict our function graph.

Properties of the function $y=x^(\frac(m)(n))$, $0 2. Is neither even nor odd.
3. Increases by $$,
b) $(2,10)$,
c) on the ray $$.
Decision.
Guys, do you remember how we found the largest and smallest value of a function on a segment in grade 10?
That's right, we used the derivative. Let's solve our example and repeat the algorithm for finding the smallest and largest value.
1. Find the derivative of the given function:
$y"=\frac(16)(5)*\frac(5)(2)x^(\frac(3)(2))-x^3=8x^(\frac(3)(2)) -x^3=8\sqrt(x^3)-x^3$.
2. The derivative exists on the entire domain of the original function, then there are no critical points. Let's find stationary points:
$y"=8\sqrt(x^3)-x^3=0$.
$8*\sqrt(x^3)=x^3$.
$64x^3=x^6$.
$x^6-64x^3=0$.
$x^3(x^3-64)=0$.
$x_1=0$ and $x_2=\sqrt(64)=4$.
Only one solution $x_2=4$ belongs to the given segment.
Let's build a table of values ​​of our function at the ends of the segment and at the extremum point:
Answer: $y_(name)=-862.65$ with $x=9$; $y_(max)=38.4$ for $x=4$.

Example. Solve the equation: $x^(\frac(4)(3))=24-x$.
Decision. The graph of the function $y=x^(\frac(4)(3))$ is increasing, while the graph of the function $y=24-x$ is decreasing. Guys, you and I know: if one function increases and the other decreases, then they intersect at only one point, that is, we have only one solution.
Note:
$8^(\frac(4)(3))=\sqrt(8^4)=(\sqrt(8))^4=2^4=16$.
$24-8=16$.
That is, for $х=8$ we got the correct equality $16=16$, this is the solution of our equation.
Answer: $x=8$.

Example.
Plot the function: $y=(x-3)^\frac(3)(4)+2$.
Decision.
The graph of our function is obtained from the graph of the function $y=x^(\frac(3)(4))$, shifting it 3 units to the right and 2 units up.

Example. Write the equation of the tangent to the line $y=x^(-\frac(4)(5))$ at the point $x=1$.
Decision. The tangent equation is determined by the formula known to us:
$y=f(a)+f"(a)(x-a)$.
In our case $a=1$.
$f(a)=f(1)=1^(-\frac(4)(5))=1$.
Let's find the derivative:
$y"=-\frac(4)(5)x^(-\frac(9)(5))$.
Let's calculate:
$f"(a)=-\frac(4)(5)*1^(-\frac(9)(5))=-\frac(4)(5)$.
Find the tangent equation:
$y=1-\frac(4)(5)(x-1)=-\frac(4)(5)x+1\frac(4)(5)$.
Answer: $y=-\frac(4)(5)x+1\frac(4)(5)$.

Tasks for independent solution

1. Find the largest and smallest value of the function: $y=x^\frac(4)(3)$ on the segment:
a) $$.
b) $(4.50)$.
c) on the ray $$.
3. Solve the equation: $x^(\frac(1)(4))=18-x$.
4. Graph the function: $y=(x+1)^(\frac(3)(2))-1$.
5. Write the equation of the tangent to the line $y=x^(-\frac(3)(7))$ at the point $x=1$.

Recall the properties and graphs of power functions with a negative integer exponent.

For even n, :

Function example:

All graphs of such functions pass through two fixed points: (1;1), (-1;1). A feature of functions of this type is their parity, the graphs are symmetrical with respect to the op-y axis.

Rice. 1. Graph of a function

For odd n, :

Function example:

All graphs of such functions pass through two fixed points: (1;1), (-1;-1). A feature of functions of this type is their oddness, the graphs are symmetrical with respect to the origin.

Rice. 2. Function Graph

Let us recall the main definition.

The degree of a non-negative number a with a rational positive exponent is called a number.

The degree of a positive number a with a rational negative exponent is called a number.

For the following equality holds:

For example: ; - the expression does not exist by definition of a degree with a negative rational exponent; exists, since the exponent is an integer,

Let us turn to the consideration of power functions with a rational negative exponent.

For example:

To plot this function, you can make a table. We will do otherwise: first, we will build and study the graph of the denominator - we know it (Figure 3).

Rice. 3. Graph of a function

The graph of the denominator function passes through a fixed point (1;1). When constructing a graph of the original function, this point remains, when the root also tends to zero, the function tends to infinity. And, conversely, as x tends to infinity, the function tends to zero (Figure 4).

Rice. 4. Function Graph

Consider one more function from the family of functions under study.

It is important that by definition

Consider the graph of the function in the denominator: , we know the graph of this function, it increases in its domain of definition and passes through the point (1; 1) (Figure 5).

Rice. 5. Function Graph

When constructing a graph of the original function, the point (1; 1) remains, when the root also tends to zero, the function tends to infinity. And, conversely, as x tends to infinity, the function tends to zero (Figure 6).

Rice. 6. Function Graph

The considered examples help to understand how the graph goes and what are the properties of the function under study - a function with a negative rational exponent.

Graphs of functions of this family pass through the point (1;1), the function decreases over the entire domain of definition.

Function scope:

The function is not bounded from above, but bounded from below. The function has neither a maximum nor a minimum value.

The function is continuous, it takes all positive values ​​from zero to plus infinity.

Convex Down Function (Figure 15.7)

Points A and B are taken on the curve, a segment is drawn through them, the entire curve is below the segment, this condition is satisfied for arbitrary two points on the curve, therefore the function is convex downward. Rice. 7.

Rice. 7. Convexity of a function

It is important to understand that the functions of this family are bounded from below by zero, but they do not have the smallest value.

Example 1 - find the maximum and minimum of the function on the interval )