Mainly review the functions and image-related knowledge learned in the past.
1. What is a function?
1. Function
A function is a rule that transforms one object into another object. The starting object is called the input, and the set from which it comes is called the domain. The returned objects are called outputs and come from a collection called the upper domain.
The essence of a function is a correspondence relationship.
For example, the following form defines a function
f is a transformation rule, and f (x) is the result of applying this transformation rule to the variable x.
A function must specify a unique output for each valid input.
The range is the set of all possible outputs
The range is actually a subset of the upper domain, which is the set of all possible outputs, while the range is the set of actual outputs.
2. Interval representation
3. Vertical line inspection method
Method used to check whether it is a function from an image perspective.
You can draw any graph you want on the coordinate plane, but the result is not necessarily the graph of a function.
Key idea: It is impossible for a function to have two points with the same x coordinate, that is, one x coordinate corresponds to two or more y coordinates.
For example, the image of a circle centered at the origin and with a radius of three units is not a function.
Because when x falls on the interval (−3, 3), for any value of x in this range, the vertical line passes through (x, 0) and intersects the circle twice.
For function, x corresponds to y, one x can correspond to one y, many x can correspond to one y, but one x cannot correspond to many y.
4. Inverse function
Given a function f, select y in the range of f. Under ideal conditions, there is only one x value that satisfies f (x) = y. (one-to-one)
If the above ideal situation holds true for every y in the range, then a new function can be defined that will reverse the transformation.
Starting from the output y, this new function finds one and only one input x that satisfies f (x) = y. This new function is called the inverse function of f.
The domain of is the same as the range of f
The range of is the same as the domain of f.
Then if there is , then there is
5. Odd and even functions
The graph of an even function has mirror symmetry about the y-axis.
The graph of odd function has point symmetry about the origin.
6. Common functions and images
graph of linear function
A function of the form f(x) = mx + b is called a linear function.
Point and slope type
如果已知直线通过点 (x0, y0), 斜率为 m, 则它的方程为 y−y0 = m(x−x0).
Find the slope
If a straight line passes through the points (x1, y1) and (x2, y2), then its slope is equal to
rational function
It is of the form p(x) / q(x), where p and q are polynomial functions, which are called rational functions. Rational functions vary in many ways, and their graphs change according to changes in the two polynomials p and q.
Exponential and logarithmic functions
Since the image of satisfies the horizontal line test, the function has an inverse function, and this inverse function is the logarithmic function with base 2
Functions with absolute values
The essential meaning of absolute value is defined on the number axis as:
|x − y|是数轴上 x 和 y 两点间的距离。Absolute value function of the form f(x) = |x|
The image of the absolute value of the function, that is, using the x-axis as a mirror, the image below the x-axis is mapped up, while the image above the x-axis remains unchanged.
END。