A power function is called a function of the form y=x n (read as y equals x to the power of n), where n is some given number. Special cases of power functions are functions of the form y=x, y=x 2, y=x 3, y=1/x and many others. Let's tell you more about each of them.
Linear function y=x 1 (y=x)
The graph is a straight line passing through the point (0;0) at an angle of 45 degrees to the positive direction of the Ox axis.
The graph is presented below.
Basic properties of a linear function:
- The function is increasing and defined on the entire number line.
- It has no maximum or minimum values.
Quadratic function y=x 2
The graph of a quadratic function is a parabola.
Basic properties of a quadratic function:
- 1. At x =0, y=0, and y>0 at x0
- 2. The quadratic function reaches its minimum value at its vertex. Ymin at x=0; It should also be noted that the function does not have a maximum value.
- 3. The function decreases on the interval (-∞;0] and increases 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 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
Let us recall the properties and graphs of power functions with a negative integer exponent.
For even n, :
Example function:
All graphs of such functions pass through two fixed points: (1;1), (-1;1). The peculiarity of functions of this type is their parity; the graphs are symmetrical relative to the op-amp axis.
Rice. 1. Graph of a function
For odd n, :
Example function:
All graphs of such functions pass through two fixed points: (1;1), (-1;-1). The peculiarity of functions of this type is that they are odd; the graphs are symmetrical with respect to the origin.
Rice. 2. Graph of a function
Let us recall the basic definition.
The power of a non-negative number a with a rational positive exponent is called a number.
The power of a positive number a with a rational negative exponent is called a number.
For the equality:
For example:
; - the expression does not exist, by definition, of a degree with a negative rational exponent; exists because the exponent is integer, 
Let's move on to considering power functions with a rational negative exponent.
For example:
To plot a graph of this function, you can create a table. We will do it differently: first we will build and study the graph of the denominator - it is known to us (Figure 3).
Rice. 3. Graph of a function
The graph of the denominator function passes through a fixed point (1;1). When plotting the graph of the original function, this point remains, while 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
Let's consider another function from the family of functions being studied.
It is important that by definition
Let's consider the graph of the function in the denominator: , the graph of this function is known to us, it increases in its domain of definition and passes through the point (1;1) (Figure 5).
Rice. 5. Graph of a function
When plotting the graph of the original function, the point (1;1) remains, while 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. Graph of a function
The considered examples help to understand how the graph flows and what are the properties of the function being studied - a function with a negative rational exponent.
The 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 limited from above, but is limited from below. The function has neither the greatest nor the least value.
The function is continuous and takes all positive values from zero to plus infinity.
The function is convex downward (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 function
It is important to understand that the functions of this family are bounded from below by zero, but do not have the smallest value.
Example 1 - find the maximum and minimum of a function on the interval)