Properties of a power function with a fractional exponent. Function

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

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Power functions, scope.

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 \u003d x ^ (\\ frac (m) (n)) $.
Consider first functions with exponent $ \\ frac (m) (n)\u003e 1 $.
Let us be given a specific function $ y \u003d x ^ 2 * 5 $.
According to the definition that we gave in the last lesson: if $ x≥0 $, that is, the domain of our function is the ray $ (x) $. Let's sketch out our function graph.

Properties of the function $ y \u003d x ^ (\\ frac (m) (n)) $, $ 0 2. It is neither even nor odd.
3. Increases by $$,
b) $ (2.10) $,
c) on the beam $$.
Decision.
Guys, do you remember how we found the largest and smallest function value on a segment in grade 10?
That's right, we used a derivative. Let's solve our example and repeat the algorithm for finding the lowest and highest value.
1. Find the derivative of the given function:
$ y "\u003d \\ frac (16) (5) * \\ frac (5) (2) x ^ (\\ frac (3) (2)) - x ^ 3 \u003d 8x ^ (\\ frac (3) (2)) -x ^ 3 \u003d 8 \\ sqrt (x ^ 3) -x ^ 3 $.
2. The derivative exists on the entire domain of the original function, then there are no critical points. Find stationary points:
$ y "\u003d 8 \\ sqrt (x ^ 3) -x ^ 3 \u003d 0 $.
$ 8 * \\ sqrt (x ^ 3) \u003d x ^ 3 $.
$ 64x ^ 3 \u003d x ^ 6 $.
$ x ^ 6-64x ^ 3 \u003d 0 $.
$ x ^ 3 (x ^ 3-64) \u003d 0 $.
$ x_1 \u003d 0 $ and $ x_2 \u003d \\ sqrt (64) \u003d 4 $.
The given segment contains only one solution $ x_2 \u003d 4 $.
Let's build a table of the values \u200b\u200bof our function at the ends of the segment and at the extremum point:
Answer: $ y_ (app.) \u003d - $ 862.65 for $ x \u003d $ 9; $ y_ (naib.) \u003d 38.4 $ for $ x \u003d 4 $.

Example. Solve the equation: $ x ^ (\\ frac (4) (3)) \u003d 24-x $.
Decision. The graph of the function $ y \u003d x ^ (\\ frac (4) (3)) $ increases, and the graph of the function $ y \u003d 24-x $ decreases. Guys, you and I know: if one function increases and the other decreases, then they intersect only at one point, that is, we have only one solution.
Note:
$ 8 ^ (\\ frac (4) (3)) \u003d \\ sqrt (8 ^ 4) \u003d (\\ sqrt (8)) ^ 4 \u003d 2 ^ 4 \u003d 16 $.
$24-8=16$.
That is, when $ x \u003d 8 $ we got the correct equality $ 16 \u003d 16 $, this is the solution to our equation.
Answer: $ x \u003d $ 8.

Example.
Plot the function: $ y \u003d (x-3) ^ \\ frac (3) (4) + 2 $.
Decision.
The graph of our function is obtained from the graph of the function $ y \u003d 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 \u003d x ^ (- \\ frac (4) (5)) $ at the point $ x \u003d 1 $.
Decision. The tangent equation is determined by the known formula:
$ y \u003d f (a) + f "(a) (x-a) $.
In our case, $ a \u003d 1 $.
$ f (a) \u003d f (1) \u003d 1 ^ (- \\ frac (4) (5)) \u003d 1 $.
Find the derivative:
$ y "\u003d - \\ frac (4) (5) x ^ (- \\ frac (9) (5)) $.
Let's calculate:
$ f "(a) \u003d - \\ frac (4) (5) * 1 ^ (- \\ frac (9) (5)) \u003d - \\ frac (4) (5) $.
Find the tangent equation:
$ y \u003d 1- \\ frac (4) (5) (x-1) \u003d - \\ frac (4) (5) x + 1 \\ frac (4) (5) $.
Answer: $ y \u003d - \\ frac (4) (5) x + 1 \\ frac (4) (5) $.

Tasks for independent solution

1. Find the largest and smallest value of the function: $ y \u003d x ^ \\ frac (4) (3) $ on the segment:
a) $$.
b) $ (4.50) $.
c) on the beam $$.
3. Solve the equation: $ x ^ (\\ frac (1) (4)) \u003d 18-x $.
4. Plot the function: $ y \u003d (x + 1) ^ (\\ frac (3) (2)) - 1 $.
5. Make the equation of the tangent to the line $ y \u003d x ^ (- \\ frac (3) (7)) $ at the point $ x \u003d 1 $.

What is a power function?

The function y \u003d x n is called a power function.

The exponent n belongs to the set of real numbers.

In the formula y \u003d x n, the argument or independent variable is x, and y is the function or dependent variable.

Power function graph

The graph of a power function when n is natural and n is greater than or equal to two is called a parabola nth degree... If n is even, then the function y \u003d x n is even, its graph is symmetric about the ordinate axis. The larger the even n, the steeper the branches of the parabola go up:

Power function with an integer negative exponent y \u003d x -n, where n is even and greater than or equal to two, is even, its graph is symmetric about the ordinate axis. Example for y \u003d x -2

Another example for y \u003d x -4:

If n is odd and n is greater than or equal to three, then the function y \u003d x n is odd, its graph is symmetric about the origin. The larger the odd n, the steeper the branches of the parabola go up:

The power function with a negative integer exponent y \u003d x -n, where n is odd and greater than or equal to three, is odd, its graph is symmetric about the origin. Example for y \u003d x -3:

The following formulas hold on the domain of definition of the power function y \u003d x p:
; ;
;
; ;
; ;
; .

Properties of power functions and their graphs

Power function with exponent equal to zero, p \u003d 0

If the exponent of the power function y \u003d x p is equal to zero, p \u003d 0, then the power function is defined for all x ≠ 0 and is constant equal to one:
y \u003d x p \u003d x 0 \u003d 1, x ≠ 0.

Power function with natural odd exponent, p \u003d n \u003d 1, 3, 5, ...

Consider a power function y \u003d x p \u003d x n with a natural odd exponent n \u003d 1, 3, 5, .... Such an indicator can also be written as: n \u003d 2k + 1, where k \u003d 0, 1, 2, 3, ... is a non-negative integer. Below are the properties and graphs of such functions.

Graph of a power function y \u003d x n with a natural odd exponent for various values \u200b\u200bof the exponent n \u003d 1, 3, 5, ....

Domain: -∞ < x < ∞
Lots of values: -∞ < y < ∞
Parity: odd, y (-x) \u003d - y (x)
Monotone: increases monotonically
Extremes: not
Convex:
at -∞< x < 0 выпукла вверх
at 0< x < ∞ выпукла вниз
Inflection points: x \u003d 0, y \u003d 0
x \u003d 0, y \u003d 0
Limits:
;
Private values:
for x \u003d -1,
y (-1) \u003d (-1) n ≡ (-1) 2k + 1 \u003d -1
for x \u003d 0, y (0) \u003d 0 n \u003d 0
for x \u003d 1, y (1) \u003d 1 n \u003d 1
Inverse function:
for n \u003d 1, the function is inverse to itself: x \u003d y
for n ≠ 1, the inverse function is a root of degree n:

Power function with a natural even exponent, p \u003d n \u003d 2, 4, 6, ...

Consider a power function y \u003d x p \u003d x n with a natural even exponent n \u003d 2, 4, 6, .... Such an indicator can also be written in the form: n \u003d 2k, where k \u003d 1, 2, 3, ... - natural. The properties and graphs of such functions are given below.

Graph of a power function y \u003d x n with a natural even exponent for various values \u200b\u200bof the exponent n \u003d 2, 4, 6, ....

Domain: -∞ < x < ∞
Lots of values: 0 ≤ y< ∞
Parity: even, y (-x) \u003d y (x)
Monotone:
for x ≤ 0 monotonically decreases
for x ≥ 0 monotonically increases
Extremes: minimum, x \u003d 0, y \u003d 0
Convex: convex down
Inflection points: not
Intersection points with coordinate axes: x \u003d 0, y \u003d 0
Limits:
;
Private values:
for x \u003d -1, y (-1) \u003d (-1) n ≡ (-1) 2k \u003d 1
for x \u003d 0, y (0) \u003d 0 n \u003d 0
for x \u003d 1, y (1) \u003d 1 n \u003d 1
Inverse function:
for n \u003d 2, square root:
for n ≠ 2, root of degree n:

Power function with negative integer exponent, p \u003d n \u003d -1, -2, -3, ...

Consider a power function y \u003d x p \u003d x n with a negative integer exponent n \u003d -1, -2, -3, .... If we put n \u003d -k, where k \u003d 1, 2, 3, ... is a natural number, then it can be represented as:

The graph of the power function y \u003d x n with a negative integer exponent for various values \u200b\u200bof the exponent n \u003d -1, -2, -3, ....

Odd exponent, n \u003d -1, -3, -5, ...

Below are the properties of the function y \u003d x n with an odd negative exponent n \u003d -1, -3, -5, ....

Domain: x ≠ 0
Lots of values: y ≠ 0
Parity: odd, y (-x) \u003d - y (x)
Monotone: decreases monotonically
Extremes: not
Convex:
at x< 0 : выпукла вверх
for x\u003e 0: convex downward
Inflection points: not
Intersection points with coordinate axes: not
Sign:
at x< 0, y < 0
for x\u003e 0, y\u003e 0
Limits:
; ; ;
Private values:
for x \u003d 1, y (1) \u003d 1 n \u003d 1
Inverse function:
for n \u003d -1,
for n< -2 ,

Even exponent, n \u003d -2, -4, -6, ...

Below are the properties of the function y \u003d x n with an even negative exponent n \u003d -2, -4, -6, ....

Domain: x ≠ 0
Lots of values: y\u003e 0
Parity: even, y (-x) \u003d y (x)
Monotone:
at x< 0 : монотонно возрастает
for x\u003e 0: monotonically decreases
Extremes: not
Convex: convex down
Inflection points: not
Intersection points with coordinate axes: not
Sign: y\u003e 0
Limits:
; ; ;
Private values:
for x \u003d 1, y (1) \u003d 1 n \u003d 1
Inverse function:
for n \u003d -2,
for n< -2 ,

Power function with rational (fractional) exponent

Consider a power function y \u003d x p with rational (fractional) exponent, where n is an integer, m\u003e 1 is a natural number. Moreover, n, m do not have common divisors.

The denominator of the fractional exponent is odd

Let the denominator of the fractional exponent be odd: m \u003d 3, 5, 7, .... In this case, the power function x p is defined for both positive and negative values \u200b\u200bof the argument x. Let us consider the properties of such power functions when the exponent p is within certain limits.

Indicator p is negative, p< 0

Let the rational exponent (with an odd denominator m \u003d 3, 5, 7, ...) be less than zero:.

Graphs of power functions with a rational negative exponent for various values \u200b\u200bof the exponent, where m \u003d 3, 5, 7, ... is odd.

Odd numerator, n \u003d -1, -3, -5, ...

We present the properties of the power function y \u003d x p with a rational negative exponent, where n \u003d -1, -3, -5, ... is an odd negative integer, m \u003d 3, 5, 7 ... is an odd natural number.

Domain: x ≠ 0
Lots of values: y ≠ 0
Parity: odd, y (-x) \u003d - y (x)
Monotone: decreases monotonically
Extremes: not
Convex:
at x< 0 : выпукла вверх
for x\u003e 0: convex downward
Inflection points: not
Intersection points with coordinate axes: not
Sign:
at x< 0, y < 0
for x\u003e 0, y\u003e 0
Limits:
; ; ;
Private values:
for x \u003d -1, y (-1) \u003d (-1) n \u003d -1
for x \u003d 1, y (1) \u003d 1 n \u003d 1
Inverse function:

Even numerator, n \u003d -2, -4, -6, ...

Properties of the power function y \u003d x p with a rational negative exponent, where n \u003d -2, -4, -6, ... is an even negative integer, m \u003d 3, 5, 7 ... is an odd natural.

Domain: x ≠ 0
Lots of values: y\u003e 0
Parity: even, y (-x) \u003d y (x)
Monotone:
at x< 0 : монотонно возрастает
for x\u003e 0: monotonically decreases
Extremes: not
Convex: convex down
Inflection points: not
Intersection points with coordinate axes: not
Sign: y\u003e 0
Limits:
; ; ;
Private values:
for x \u003d -1, y (-1) \u003d (-1) n \u003d 1
for x \u003d 1, y (1) \u003d 1 n \u003d 1
Inverse function:

The exponent p is positive, less than one, 0< p < 1

The graph of a power function with a rational exponent (0< p < 1 ) при различных значениях показателя степени , где m = 3, 5, 7, ... - нечетное.

Odd numerator, n \u003d 1, 3, 5, ...

< p < 1 , где n = 1, 3, 5, ... - нечетное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

Domain: -∞ < x < +∞
Lots of values: -∞ < y < +∞
Parity: odd, y (-x) \u003d - y (x)
Monotone: increases monotonically
Extremes: not
Convex:
at x< 0 : выпукла вниз
for x\u003e 0: convex upward
Inflection points: x \u003d 0, y \u003d 0
Intersection points with coordinate axes: x \u003d 0, y \u003d 0
Sign:
at x< 0, y < 0
for x\u003e 0, y\u003e 0
Limits:
;
Private values:
for x \u003d -1, y (-1) \u003d -1
for x \u003d 0, y (0) \u003d 0
for x \u003d 1, y (1) \u003d 1
Inverse function:

Even numerator, n \u003d 2, 4, 6, ...

The properties of a power function y \u003d x p with a rational exponent within 0< p < 1 , где n = 2, 4, 6, ... - четное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

Domain: -∞ < x < +∞
Lots of values: 0 ≤ y< +∞
Parity: even, y (-x) \u003d y (x)
Monotone:
at x< 0 : монотонно убывает
for x\u003e 0: monotonically increases
Extremes: minimum at x \u003d 0, y \u003d 0
Convex: is convex upward for x ≠ 0
Inflection points: not
Intersection points with coordinate axes: x \u003d 0, y \u003d 0
Sign: for x ≠ 0, y\u003e 0
Limits:
;
Private values:
for x \u003d -1, y (-1) \u003d 1
for x \u003d 0, y (0) \u003d 0
for x \u003d 1, y (1) \u003d 1
Inverse function:

P is greater than one, p\u003e 1

A graph of a power function with a rational exponent (p\u003e 1) for different values \u200b\u200bof the exponent, where m \u003d 3, 5, 7, ... is odd.

Odd numerator, n \u003d 5, 7, 9, ...

Properties of the power function y \u003d x p with a rational exponent greater than one:. Where n \u003d 5, 7, 9, ... is an odd natural, m \u003d 3, 5, 7 ... is an odd natural.

Domain: -∞ < x < ∞
Lots of values: -∞ < y < ∞
Parity: odd, y (-x) \u003d - y (x)
Monotone: increases monotonically
Extremes: not
Convex:
at -∞< x < 0 выпукла вверх
at 0< x < ∞ выпукла вниз
Inflection points: x \u003d 0, y \u003d 0
Intersection points with coordinate axes: x \u003d 0, y \u003d 0
Limits:
;
Private values:
for x \u003d -1, y (-1) \u003d -1
for x \u003d 0, y (0) \u003d 0
for x \u003d 1, y (1) \u003d 1
Inverse function:

Even numerator, n \u003d 4, 6, 8, ...

Properties of the power function y \u003d x p with a rational exponent greater than one:. Where n \u003d 4, 6, 8, ... is an even natural, m \u003d 3, 5, 7 ... is an odd natural.

Domain: -∞ < x < ∞
Lots of values: 0 ≤ y< ∞
Parity: even, y (-x) \u003d y (x)
Monotone:
at x< 0 монотонно убывает
for x\u003e 0 monotonically increases
Extremes: minimum at x \u003d 0, y \u003d 0
Convex: convex down
Inflection points: not
Intersection points with coordinate axes: x \u003d 0, y \u003d 0
Limits:
;
Private values:
for x \u003d -1, y (-1) \u003d 1
for x \u003d 0, y (0) \u003d 0
for x \u003d 1, y (1) \u003d 1
Inverse function:

The denominator of the fractional exponent is even

Let the denominator of the fractional exponent be even: m \u003d 2, 4, 6, .... In this case, the exponential function x p is undefined for negative argument values. Its properties are the same as those of a power function with an irrational exponent (see the next section).

Power function with irrational exponent

Consider a power function y \u003d x p with an irrational exponent p. The properties of such functions differ from those considered above in that they are not defined for negative values \u200b\u200bof the argument x. For positive values \u200b\u200bof the argument, the properties depend only on the magnitude of the exponent p and do not depend on whether p is integer, rational, or irrational.

y \u003d x p for different values \u200b\u200bof the exponent p.

Power function with negative exponent p< 0

Domain: x\u003e 0
Lots of values: y\u003e 0
Monotone: decreases monotonically
Convex: convex down
Inflection points: not
Intersection points with coordinate axes: not
Limits: ;
Private value: For x \u003d 1, y (1) \u003d 1 p \u003d 1

Power function with positive exponent p\u003e 0

Indicator less than one 0< p < 1

Domain: x ≥ 0
Lots of values: y ≥ 0
Monotone: increases monotonically
Convex: convex up
Inflection points: not
Intersection points with coordinate axes: x \u003d 0, y \u003d 0
Limits:
Private values: For x \u003d 0, y (0) \u003d 0 p \u003d 0.
For x \u003d 1, y (1) \u003d 1 p \u003d 1

Indicator greater than one p\u003e 1

Domain: x ≥ 0
Lots of values: y ≥ 0
Monotone: increases monotonically
Convex: convex down
Inflection points: not
Intersection points with coordinate axes: x \u003d 0, y \u003d 0
Limits:
Private values: For x \u003d 0, y (0) \u003d 0 p \u003d 0.
For x \u003d 1, y (1) \u003d 1 p \u003d 1

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Technical Institutions, "Lan", 2009.

Knowledge basic elementary functions, their properties and graphs no less important than knowing the multiplication table. They are like a foundation, everything is based on them, everything is built from them and everything comes down to them.

In this article we will list all the basic elementary functions, give their graphs and give them without derivation and proof. properties of basic elementary functions according to the scheme:

• the behavior of the function at the boundaries of the domain of definition, vertical asymptotes (if necessary, see the article on the classification of break points of a function);
• even and odd;
• intervals of convexity (upward convexity) and concavity (downward convexity), inflection points (if necessary, see the article convexity of a function, convexity direction, inflection points, convexity and inflection conditions);
• oblique and horizontal asymptotes;
• special points of functions;
• special properties some functions (for example, the smallest positive period for trigonometric functions).

If you are interested in or, then you can go to these sections of the theory.

Basic Elementary Functions are: constant function (constant), n-th root, power function, exponential, logarithmic function, trigonometric and inverse trigonometric functions.

Page navigation.

Permanent function.

A constant function is defined on the set of all real numbers by the formula, where C is some real number. The constant function assigns to each real value of the independent variable x the same value of the dependent variable y - the value of C. A constant function is also called a constant.

The graph of a constant function is a straight line parallel to the abscissa axis and passing through a point with coordinates (0, C). As an example, we will show the graphs of constant functions y \u003d 5, y \u003d -2 and, to which the black, red and blue lines correspond in the figure below, respectively.

Constant function properties.

• Domain: the whole set of real numbers.
• The constant function is even.
• Range of values: a set consisting of singular FROM .
• The constant function is non-increasing and non-decreasing (that's why it is constant).
• It makes no sense to talk about the convexity and concavity of a constant.
• There are no asymptotes.
• The function passes through the point (0, C) of the coordinate plane.

Nth root.

Consider the basic elementary function, which is given by the formula, where n is a natural number greater than one.

Nth root, n is an even number.

Let's start with the function n-th root for even values \u200b\u200bof the exponent of the root n.

For example, we give a figure with images of graphs of functions and, they correspond to black, red and blue lines.

The graphs of functions of the root of an even degree have a similar form for other values \u200b\u200bof the indicator.

Properties of the nth root function for even n.

Nth root, n is an odd number.

The nth root function with odd root exponent n is defined on the entire set of real numbers. For example, we will give graphs of functions and, they correspond to the black, red and blue curves.

For other odd values \u200b\u200bof the root exponent, the graphs of the function will have a similar appearance.

Properties of the nth root function for odd n.

Power function.

The power function is given by a formula of the form.

Consider the form of the power function graphs and the properties of the power function depending on the value of the exponent.

Let's start with a power function with integer exponent a. In this case, the form of graphs of power functions and the properties of the functions depend on the evenness or oddness of the exponent, as well as on its sign. Therefore, we first consider power functions for odd positive values \u200b\u200bof the exponent a, then for even positive exponents, then for odd negative exponents, and, finally, for even negative a.

The properties of power functions with fractional and irrational exponents (as well as the form of graphs of such power functions) depend on the value of the exponent a. They will be considered, firstly, for a from zero to one, secondly, when a is large units, thirdly, when a is from minus one to zero, and fourthly, when a is less than minus one.

To conclude this section, for completeness, we describe a power function with zero exponent.

Power function with an odd positive exponent.

Consider a power function with an odd positive exponent, that is, with a \u003d 1,3,5,….

The figure below shows graphs of power-law functions - black line, - blue line, - red line, - green line. For a \u003d 1 we have linear function y \u003d x.

Properties of a power function with an odd positive exponent.

Power function with even positive exponent.

Consider a power function with an even positive exponent, that is, with a \u003d 2,4,6,….

As an example, we will give graphs of power functions - black line, - blue line, - red line. For a \u003d 2 we have a quadratic function whose graph is quadratic parabola.

Properties of a power function with an even positive exponent.

Power function with odd negative exponent.

Look at the graphs of the power function for odd negative values \u200b\u200bof the exponent, that is, for a \u003d -1, -3, -5,….

The figure shows the graphs of power functions as examples - black line, - blue line, - red line, - green line. For a \u003d -1 we have inverse proportionwhose graph is hyperbola.

Properties of a power function with an odd negative exponent.

Power function with even negative exponent.

Let us pass to the power function for a \u003d -2, -4, -6,….

The figure shows graphs of power functions - black line, - blue line, - red line.

Properties of a power function with an even negative exponent.

Power function with rational or irrational exponent, the value of which is greater than zero and less than one.

Note! If a is a positive fraction with an odd denominator, then some authors consider the interval to be the domain of definition of the power function. At the same time, it is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and the principles of analysis DO NOT DETERMINE power functions with an exponent in the form of a fraction with an odd denominator for negative values \u200b\u200bof the argument. We will adhere to just such a view, that is, we will consider the domains of definition of power functions with fractional positive exponents of the set. We encourage students to get to know your teacher's perspective on this delicate point to avoid controversy.

Consider a power function with rational or irrational exponent a, and.

Here are the graphs of power functions for a \u003d 11/12 (black line), a \u003d 5/7 (red line), (blue line), a \u003d 2/5 (green line).

Power function with non-integer rational or irrational exponent greater than one.

Consider a power function with noninteger rational or irrational exponent a, and.

Let us present the graphs of power functions given by the formulas (black, red, blue and green lines, respectively).

>

For other values \u200b\u200bof the exponent a, the graphs of the function will look similar.

Power function properties for.

Power function with real exponent greater than minus one and less than zero.

Note! If a is a negative fraction with an odd denominator, then some authors consider the interval ... At the same time, it is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and the principles of analysis DO NOT DETERMINE power functions with an exponent in the form of a fraction with an odd denominator for negative values \u200b\u200bof the argument. We will adhere to just such a view, that is, we will consider the domains of definition of power functions with fractional fractional negative exponents, respectively. We encourage students to get to know your teacher's perspective on this delicate point in order to avoid disagreement.

We pass to the power function, kg.

In order to have a good idea of \u200b\u200bthe form of graphs of power functions for, we give examples of graphs of functions (black, red, blue and green curves, respectively).

Properties of a power function with exponent a,.

Power function with a non-integer real exponent less than minus one.

Let us give examples of graphs of power functions for , they are depicted with black, red, blue and green lines, respectively.

Properties of a power function with a non-integer negative exponent less than minus one.

With a \u003d 0 and we have a function - this is a straight line from which the point (0; 1) is excluded (the expression 0 0 was agreed not to attach any meaning).

Exponential function.

One of the basic elementary functions is the exponential function.

The exponential function graph, where and takes on a different form depending on the value of the base a. Let's figure it out.

First, consider the case when the base of the exponential function takes a value from zero to one, that is,.

For example, we will give graphs of the exponential function with a \u003d 1/2 - the blue line, a \u003d 5/6 - the red line. The plots of the exponential function with other values \u200b\u200bof the base from the interval have a similar form.

Properties of exponential function with base less than one.

We turn to the case when the base of the exponential function is greater than one, that is,.

As an illustration, we present graphs of exponential functions - blue line and - red line. For other values \u200b\u200bof the base, greater than one, the graphs of the exponential function will have a similar appearance.

Properties of the exponential function with a base greater than one.

Logarithmic function.

The next main elementary function is the logarithmic function, where,. The logarithmic function is defined only for positive values \u200b\u200bof the argument, that is, for.

The graph of the logarithmic function takes on a different form depending on the value of the base a.

Properties of power functions and their graphs

Power function with exponent equal to zero, p \u003d 0

If the exponent of the power function y \u003d x p is equal to zero, p \u003d 0, then the power function is defined for all x ≠ 0 and is constant equal to one:
y \u003d x p \u003d x 0 \u003d 1, x ≠ 0.

Power function with natural odd exponent, p \u003d n \u003d 1, 3, 5, ...

Consider a power function y \u003d xp \u003d xn with a natural odd exponent n \u003d 1, 3, 5, .... This exponent can also be written as: n \u003d 2k + 1, where k \u003d 0, 1, 2, 3,. .. is not a negative whole. Below are the properties and graphs of such functions.

Graph of a power function y \u003d x n with a natural odd exponent for various values \u200b\u200bof the exponent n \u003d 1, 3, 5, ....

Range of definition: –∞< x < ∞

Range of values: –∞< y < ∞

Extremes: no

Convex:

at –∞< x < 0 выпукла вверх

at 0< x < ∞ выпукла вниз

Inflection points: x \u003d 0, y \u003d 0

Private values:

for x \u003d –1, y (–1) \u003d (–1) n ≡ (–1) 2m + 1 \u003d –1

for x \u003d 0, y (0) \u003d 0 n \u003d 0

for x \u003d 1, y (1) \u003d 1 n \u003d 1

Power function with a natural even exponent, p \u003d n \u003d 2, 4, 6, ...

Consider a power function y \u003d xp \u003d xn with a natural even exponent n \u003d 2, 4, 6, .... Such an exponent can also be written as: n \u003d 2k, where k \u003d 1, 2, 3, ... is a natural ... The properties and graphs of such functions are given below.

Graph of a power function y \u003d x n with a natural even exponent for various values \u200b\u200bof the exponent n \u003d 2, 4, 6, ....

Range of definition: –∞< x < ∞

Range of values: 0 ≤ y< ∞

Monotone:

at x< 0 монотонно убывает

for x\u003e 0 monotonically increases

Extrema: minimum, x \u003d 0, y \u003d 0

Bulge: convex down

Inflection points: no

Intersection points with coordinate axes: x \u003d 0, y \u003d 0
Private values:

for x \u003d –1, y (–1) \u003d (–1) n ≡ (–1) 2m \u003d 1

for x \u003d 0, y (0) \u003d 0 n \u003d 0

for x \u003d 1, y (1) \u003d 1 n \u003d 1

Power function with negative integer exponent, p \u003d n \u003d -1, -2, -3, ...

Consider a power function y \u003d xp \u003d xn with a negative integer exponent n \u003d -1, -2, -3, .... If we put n \u003d –k, where k \u003d 1, 2, 3, ... is a natural number, then it can be represented as:

The graph of the power function y \u003d x n with an integer negative exponent for various values \u200b\u200bof the exponent n \u003d -1, -2, -3, ....

Odd exponent, n \u003d -1, -3, -5, ...

Below are the properties of the function y \u003d x n with an odd negative exponent n \u003d -1, -3, -5, ....

Domain: x ≠ 0

Set of values: y ≠ 0

Parity: odd, y (–x) \u003d - y (x)

Extremes: no

Convex:

at x< 0: выпукла вверх

for x\u003e 0: convex downward

Inflection points: no

Sign: for x< 0, y < 0

for x\u003e 0, y\u003e 0

Private values:

for x \u003d 1, y (1) \u003d 1 n \u003d 1

Even exponent, n \u003d -2, -4, -6, ...

Below are the properties of the function y \u003d x n with an even negative exponent n \u003d -2, -4, -6, ....

Domain: x ≠ 0

Set of values: y\u003e 0

Parity: even, y (–x) \u003d y (x)

Monotone:

at x< 0: монотонно возрастает

for x\u003e 0: monotonically decreases

Extremes: no

Bulge: convex down

Inflection points: no

Intersection points with coordinate axes: none

Sign: y\u003e 0

Private values:

for x \u003d –1, y (–1) \u003d (–1) n \u003d 1

for x \u003d 1, y (1) \u003d 1 n \u003d 1

Power function with rational (fractional) exponent

Consider a power function y \u003d x p with a rational (fractional) exponent, where n is an integer and m\u003e 1 is a natural number. Moreover, n, m do not have common divisors.

The denominator of the fractional exponent is odd

Let the denominator of the fractional exponent be odd: m \u003d 3, 5, 7, .... In this case, the power function x p is defined for both positive and negative values \u200b\u200bof the argument. Let us consider the properties of such power functions when the exponent p is within certain limits.

Indicator p is negative, p< 0

Let the rational exponent (with an odd denominator m \u003d 3, 5, 7, ...) be less than zero: .

Power function graphs with a rational negative exponent for various values \u200b\u200bof the exponent, where m \u003d 3, 5, 7, ... is odd.

Odd numerator, n \u003d -1, -3, -5, ...

We present the properties of a power function y \u003d x p with a rational negative exponent, where n \u003d -1, -3, -5, ... is an odd negative integer, m \u003d 3, 5, 7 ... is an odd positive integer.

Domain: x ≠ 0

Set of values: y ≠ 0

Parity: odd, y (–x) \u003d - y (x)

Monotonicity: decreases monotonically

Extremes: no

Convex:

at x< 0: выпукла вверх

for x\u003e 0: convex downward

Inflection points: no

Intersection points with coordinate axes: none

at x< 0, y < 0

for x\u003e 0, y\u003e 0

Private values:

at x \u003d –1, y (–1) \u003d (–1) n \u003d –1

for x \u003d 1, y (1) \u003d 1 n \u003d 1

Even numerator, n \u003d -2, -4, -6, ...

Properties of the power function y \u003d x p with a rational negative exponent, where n \u003d -2, -4, -6, ... is an even negative integer, m \u003d 3, 5, 7 ... is an odd natural.

Domain: x ≠ 0

Set of values: y\u003e 0

Parity: even, y (–x) \u003d y (x)

Monotone:

at x< 0: монотонно возрастает

for x\u003e 0: monotonically decreases

Extremes: no

Bulge: convex down

Inflection points: no

Intersection points with coordinate axes: none

Sign: y\u003e 0

The exponent p is positive, less than one, 0< p < 1

Power function graph with a rational exponent (0< p < 1) при различных значениях показателя степени , где m = 3, 5, 7, ... - нечетное.

Odd numerator, n \u003d 1, 3, 5, ...

< p < 1, где n = 1, 3, 5, ... - нечетное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

Range of definition: –∞< x < +∞

Range of values: –∞< y < +∞

Parity: odd, y (–x) \u003d - y (x)

Monotony: monotonically increasing

Extremes: no

Convex:

at x< 0: выпукла вниз

for x\u003e 0: convex upward

Inflection points: x \u003d 0, y \u003d 0

Intersection points with coordinate axes: x \u003d 0, y \u003d 0

at x< 0, y < 0

for x\u003e 0, y\u003e 0

Private values:

at x \u003d –1, y (–1) \u003d –1

for x \u003d 0, y (0) \u003d 0

for x \u003d 1, y (1) \u003d 1

Even numerator, n \u003d 2, 4, 6, ...

The properties of a power function y \u003d x p with a rational exponent within 0< p < 1, где n = 2, 4, 6, ... - четное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

Range of definition: –∞< x < +∞

Range of values: 0 ≤ y< +∞

Parity: even, y (–x) \u003d y (x)

Monotone:

at x< 0: монотонно убывает

for x\u003e 0: monotonically increases

Extrema: minimum at x \u003d 0, y \u003d 0

Convexity: convex upward for x ≠ 0

Inflection points: no

Intersection points with coordinate axes: x \u003d 0, y \u003d 0

Sign: for x ≠ 0, y\u003e 0