Generally speaking, if X and Y correspond to a certain correspondence f(x) and y=f(x), is the inverse function of y=f(x) x=f(y) or y=f? ㈩. The condition for the existence of the inverse function (single-valued function by default) is that the original function must be in one-to-one correspondence (not necessarily in the whole number domain). Note: superscript "? 1 "does not represent a power.
The images of (1) function f(x) and its inverse function f- 1(x) are symmetric about the straight line y=x;
(2) The [necessary and sufficient condition] that a function has an inverse function is that the [domain] and [range] of the function are [one-to-one mapping;
(3) The function and its inverse function are monotonous in the corresponding interval;
(4) Most [even-numbered functions] have no inverse function (when the function y=f(x), the domain is {0}, and f(x)=C (where c is a constant), the function f(x) is even and has an inverse function, and the domain of the inverse function is {C} and the range is {0}). [odd function] There is not necessarily an inverse function. When it is cut by a straight line perpendicular to the Y axis, it can pass through two or more points, that is, there is no inverse function. If a [odd function] has an inverse function, its inverse function is also odd function.
(5) The monotonicity of continuous functions is consistent in the corresponding interval;
(6) The strict increase (decrease) function must have the inverse function of strict increase (decrease);
(7) Inverse functions are mutually unique;
(8) The corresponding rules of opposites between [domain] and [scope] are reciprocal (three opposites);
(9) the [derivative] relation of the inverse function: if x=f(y) is strictly monotonic and differentiable in the open interval I, and f'(y)≠0, then its inverse function y=f-1 (x) is in the interval S = {x | x = f (y), y.
(10) The inverse function of y = x is itself.
Basic elementary functions include power function, exponential function, logarithmic function, trigonometric function, inverse trigonometric function and constant function.
Generally speaking, a function in the form of y=x α (α is a rational number), that is, a function with the base as the independent variable, the power as the dependent variable and the exponent as the constant is called a power function. For example, functions y = x 0, y = x1,y = x 2, y=x-1 (Note: x = x-1=1/x y = x 0,
The image of power function must be in the first quadrant, not in the fourth quadrant. Whether it is in the second quadrant or the third quadrant depends on the parity of the function. The power function image can only be in two quadrants at the same time; If the power function image intersects the coordinate axis, the intersection must be the origin.
When α >; 0, the power function y=x α has the following properties:
A, the images all pass through the point (1,1) (0,0);
B, the function in the image is the increasing function in the interval [0, +∞);
C, in the first quadrant, α >; 1, the derivative value increases gradually; When α= 1, the derivative is constant; 0 & ltα& lt； 1, the derivative value gradually decreases and approaches 0;
When α
A, the images all pass through the point (1,1);
B, the image is a decreasing function in the interval (0, +∞); (Content supplement: If it is X -2, it is easy to get that it is an even function. Using symmetry, the symmetry axis is the Y axis, and the image can be monotonically increased in the interval (-∞, 0). The same is true for other even functions)
C, in the first quadrant, there are two asymptotes (coordinate axes), the independent variable approaches 0, the function value approaches +∞, the independent variable approaches +∞, and the function value approaches 0.
When α=0, the power function y=x a has the following properties:
An image with a and y = x 0 is a straight line with y= 1 minus a point (0, 1). Its image is not a straight line.
Generally speaking, the function of y = a x (a is constant and a >；; 0, a≠ 1) is called exponential function, and the domain of the function is r, and the range of values for all exponential functions is (0, +∞).
Exponential function is an important function in mathematics. This function applied to the value e is written as exp( x). It can also be equivalently written as e x, where e is a mathematical constant and the base of natural logarithm, which is approximately equal to 2.7 1828 1828, also known as Euler number.
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Generally speaking, logarithmic function is a function with power (true number) as independent variable, exponent as dependent variable and base as constant.
Logarithmic function is one of the six basic elementary functions. Where the logarithm is defined as:
If ax = n (a > 0, and a≠ 1), then the number x is called the logarithm of the base of n, denoted as x=log a N, and read as the logarithm of the base of n, where a is called the base of logarithm and n is called a real number.
In general, the function y = logax(a >；; 0, and a≠ 1) is called logarithmic function, that is, a function with power (real number) as independent variable, exponent as dependent variable and base constant as constant is called logarithmic function.
Where x is the independent variable and the domain of the function is (0, +∞), that is, x >；; 0。 It is actually the inverse function of exponential function, which can be expressed as x=ay. Therefore, the stipulation of a in exponential function is also applicable to logarithmic function.
Sine function: y=sin x
Symmetry axis: x=kπ+π/2(k∈Z)
Center of symmetry: (kπ, 0)(k∈Z)
Cosine function: y=cos x
Symmetry axis: x=kπ(k∈Z)
Center of symmetry: kπ+π/2,0) (k ∈ z)
Tangent function: y=tan x
Symmetry axis: none
Center of symmetry: (kπ/2+π/2,0) (k ∈ z)
Cotangent function: y=cot x
Symmetry axis: none
Center of symmetry: (kπ/2,0) (k ∈ z)
Secant function: y=sec x
Symmetry axis: x=kπ(k∈Z)
Center of symmetry: (kπ+π/2,0) (k ∈ z)
Cotangent function: y=csc x
Symmetry axis: x=kπ+π/2(k∈Z)
Center of symmetry: (kπ, 0)(k∈Z)
Inverse trigonometric function is a basic elementary function. It is a general term for functions such as arcsine x, arccosine arccos x, arctangent arctan x, arccot x, arcsec x, arccsc x, which respectively represent the angles of arcsine, arccosine, arctangent, anti-cotangent, arctangent and anti-cotangent.
It can't be understood as the inverse function of trigonometric function in a narrow sense, but it is a multi-valued function. The inverse function of trigonometric function is not a single-valued function, because it does not meet the requirement that the independent variable corresponds to a function value, and its image is symmetrical with its original function about function y = X. Euler put forward the concept of inverse trigonometric function, and expressed it in the form of "arc+function name" for the first time.
In order to define the inverse trigonometric function as a single-valued function, the value y of the arcsine function is defined as -π/2≤y≤π/2, and y is taken as the principal value of the arcsine function, which is denoted as y = arcsinx. Accordingly, the principal value of the inverse cosine function y=arccos x is limited to 0 ≤ y ≤π; The principal value of arctangent function y=arctan x is limited to -π/2.
The inverse function of sine function y=sin x on [-π/2, π/2] is called arcsine function. Arcsinx represents an angle with a sine value of x, and the value range of the angle is within the range of [-π/2, π/2]. Domain [- 1, 1], range [-π/2, π/2]. [ 1]
The inverse function of cosine function y=cos x on [0, π] is called anti-cosine function. Recorded as arccosx, it represents the angle whose cosine value is x, and the range of the angle is within the range of [0, π]. Definition domain [- 1, 1] and value domain [0, π]. [ 1]
The inverse function of tangent function y=tan x on (-π/2, π/2) is called arc tangent function. Recorded as arctanx, it represents the angle with the tangent value of x, and the value range of the angle is within the range of (-π/2, π/2). The field r, range (-π/2, π/2). [ 1]
The inverse function of cotangent function y=cot x on (0, π) is called inverse cotangent function. Write arccotx
That represents the angle with cotangent value of x, and the range of the angle is within the range of (0, π). Definition domain r, value domain (0, π). [ 1]
The inverse function of secx = y on [0, π/2] u (π/2, π) is called arc tangent function. Arcsecx, which represents the angle with secant value of x, is in the range of [0, π/2] u (π/2, π). The definition domain (-∞,-1)U[ 1, +∞) and the value domain [0, π/2] u (π/2, π). [ 1]
The inverse function of cotangent function y=csc x on [-π/2,0) u (0,π/2] is called the inverse cotangent function. Arccscx represents an angle whose cotangent value is x, and the range of the angle is [-π/2,0) u (0,π/2]. The definition domain (-∞,-1)U[ 1, +∞) and the value domain [-π/2,0 0] u (0 0,π/2]. [ 1]
In mathematics, a constant function (also called a constant function) refers to a function whose value is constant (that is, constant). For example, we have the function f(x)=4, because f maps any value to 4, so f is a constant. More generally, for a function f: A→B, if f(x)=f(y) exists for all x and y in a, then f is a constant function.