Power function with natural even exponent. Power function
Function where X– variable quantity, A– a given number is called Power function .
If then is a linear function, its graph is a straight line (see paragraph 4.3, Fig. 4.7).
If then is a quadratic function, its graph is a parabola (see paragraph 4.3, Fig. 4.8).
If then its graph is a cubic parabola (see paragraph 4.3, Fig. 4.9).
Power function
This is the inverse function for
1. Domain: 
2. Multiple meanings:
3. Even and odd: function is odd.
4. Function frequency: non-periodic.
5. Function zeros: X= 0 – the only zero.
6. The function does not have a maximum or minimum value.
7.
8. Graph of a function Symmetrical to the graph of a cubic parabola relative to a straight line Y=X and is shown in Fig. 5.1.
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Power function
1. Domain: 
2. Multiple meanings:
3. Even and odd: the function is even.
4. Function frequency: non-periodic.
5. Function zeros: single zero X = 0.
6. The largest and smallest values of the function: takes the smallest value for X= 0, it is equal to 0.
7. Increasing and decreasing intervals: the function is decreasing on the interval and increasing on the interval
8. Graph of a function(for each N Î N) is “similar” to the graph of a quadratic parabola (function graphs are shown in Fig. 5.2).
Power function
1. Domain:
2. Multiple meanings: 
3. Even and odd: function is odd.
4. Function frequency: non-periodic.
5. Function zeros: X= 0 – the only zero.
6. Highest and lowest values:
7. Increasing and decreasing intervals: the function is increasing over the entire domain of definition.
8. Graph of a function(for each ) is “similar” to the graph of a cubic parabola (function graphs are shown in Fig. 5.3).
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Power function
1. Domain:
2. Multiple meanings:
3. Even and odd: function is odd.
4. Function frequency: non-periodic.
5. Function zeros: has no zeros.
6. The largest and smallest values of the function: the function does not have the largest and smallest values for any
7. Increasing and decreasing intervals: the function is decreasing in its domain of definition.
8. Asymptotes:(axis OU) – vertical asymptote;
(axis Oh) – horizontal asymptote.
9. Graph of a function(for anyone N) is “similar” to the graph of a hyperbola (function graphs are shown in Fig. 5.4).
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Power function
1. Domain:
2. Multiple meanings:
3. Even and odd: the function is even.
4. Function frequency: non-periodic.
5. The largest and smallest values of the function: the function does not have the largest and smallest values for any
6. Increasing and decreasing intervals: the function is increasing by and decreasing by
7. Asymptotes: X= 0 (axis OU) – vertical asymptote;
Y= 0 (axis Oh) – horizontal asymptote.
8. Function graphs They are quadratic hyperbolas (Fig. 5.5).
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Power function
1. Domain:
2. Multiple meanings:
3. Even and odd: the function does not have the property of even and odd.
4. Function frequency: non-periodic.
5. Function zeros: X= 0 – the only zero.
6. The largest and smallest values of the function: the function takes the smallest value equal to 0 at the point X= 0; doesn't matter the most.
7. Increasing and decreasing intervals: the function is increasing over the entire domain of definition.
8. Each such function for a certain exponent is the inverse of the function provided
9. Graph of a function"resembles" the graph of a function for any N and is shown in Fig. 5.6.
Power function
1. Domain:
2. Multiple meanings:
3. Even and odd: function is odd.
4. Function frequency: non-periodic.
5. Function zeros: X= 0 – the only zero.
6. The largest and smallest values of the function: the function does not have the largest and smallest values for any
7. Increasing and decreasing intervals: the function is increasing over the entire domain of definition.
8. Graph of a function Shown in Fig. 5.7.
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For the convenience of considering a power function, we will consider 4 separate cases: a power function with a natural exponent, a power function with an integer exponent, a power function with a rational exponent, and a power function with an irrational exponent.
Power function with natural exponent
First, let's introduce the concept of a degree with a natural exponent.
Definition 1
The power of a real number $a$ with natural exponent $n$ is a number equal to the product of $n$ factors, each of which equals the number $a$.
Picture 1.
$a$ is the base of the degree.
$n$ is the exponent.
Let us now consider a power function with a natural exponent, its properties and graph.
Definition 2
$f\left(x\right)=x^n$ ($n\in N)$ is called a power function with a natural exponent.
For further convenience, we consider separately a power function with an even exponent $f\left(x\right)=x^(2n)$ and a power function with an odd exponent $f\left(x\right)=x^(2n-1)$ ($n\in N)$.
Properties of a power function with a natural even exponent
$f\left(-x\right)=((-x))^(2n)=x^(2n)=f(x)$ -- the function is even.
Value area -- $\
The function decreases as $x\in (-\infty ,0)$ and increases as $x\in (0,+\infty)$.
$f("")\left(x\right)=(\left(2n\cdot x^(2n-1)\right))"=2n(2n-1)\cdot x^(2(n-1 ))\ge 0$
The function is convex over the entire domain of definition.
Behavior at the ends of the domain:
\[(\mathop(lim)_(x\to -\infty ) x^(2n)\ )=+\infty \] \[(\mathop(lim)_(x\to +\infty ) x^( 2n)\ )=+\infty \]
Graph (Fig. 2).
Figure 2. Graph of the function $f\left(x\right)=x^(2n)$
Properties of a power function with a natural odd exponent
The domain of definition is all real numbers.
$f\left(-x\right)=((-x))^(2n-1)=(-x)^(2n)=-f(x)$ -- the function is odd.
$f(x)$ is continuous over 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
First, let's introduce the concept of a degree with an integer exponent.
Definition 3
The power of a real number $a$ with integer exponent $n$ is determined by the formula:
Figure 4.
Let us now consider 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 an 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 discussed it above. For $n=0$ we get a linear function $y=1$. We will 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 domain of definition 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 over the entire domain of definition.
Scope:
If the exponent is even, then $(0,+\infty)$; if it is odd, then $\left(-\infty ,0\right)(0,+\infty)$.
For an odd exponent, the function decreases as $x\in \left(-\infty ,0\right)(0,+\infty)$. If the exponent is even, 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 of definition