In mathematics, the graph (or image) of function f refers to the set composed of all ordered pairs (x, f(x)) [1]. Specifically, if x is a real number, then the function diagram is represented as a curve in the plane rectangular coordinate system. If the function independent variable x is an ordered pair (x 1, x2) composed of two real numbers, then the graph is a set composed of all triple orders (x 1, x2, f(x 1, x2)) and presents as a curved surface (see 3D computer graphics).
Geometry and Equation Theory of Function Images
Function image (function image)
Point set {(x, y)èy = x}
The image of a linear function is called the image of a function y = x.
The independent variable x and the dependent variable y of a linear function have the following relationship:
Y=kx+b(k, b is a constant, k≠0) is called a linear function of x.
In particular, when b=0, y is a proportional function of x.
If the relationship between two variables X and Y can be expressed in the form of y=kx+b(k, b is a constant, k≠0), then Y is a linear function of X (X is an independent variable and Y is a dependent variable). In particular, when b=0, y is said to be a proportional function of x.
Image properties:
1. Practice and graphics: Through the following three steps.
(1) Calculate the intersection coordinates of the function image with the Y axis and the X axis.
(2) tracking points;
(3) Connecting lines can make images of linear functions-straight lines.
2. Property: any point P(x, y) on the linear function satisfies the equation: y = kx+b.
3. Quadrant where k, b and function images are located.
When k>0, the straight line must pass through the first and third quadrants. From left to right, Y increases with the increase of X;
When k < 0, the straight line must pass through the second and fourth quadrants, and from left to right, y decreases with the increase of x;
When b>0, the straight line must pass through the first and second quadrants; When b<0, the straight line must pass through three or four quadrants.
Especially, when b=O, the straight line passing through the origin o (0 0,0) represents the image of the proportional function.
At this time, when k>0, the straight line only passes through the first and third quadrants; When k < 0, the straight line only passes through the second and fourth quadrants.
4. The set of values of independent variables in (1) function relation is called the domain of function. Finding the domain of a function represented by an analytical expression is to find the intersection of sets that make each component of the function meaningful. For the definition domain of functional relationship in practical problems, we need to consider the conditions of practical problems.
(2) The set formed by the range and the function values corresponding to all x values in the defined range is called the range of the function.
(3) Monotonicity definition: For the function f(x) on a given interval.