"Function and its Representation Method" Concept and Properties of Function PPT (Concept of Function in Lesson 1)

Description

"Function and its Representation Method" Concept and Properties of Function PPT (Concept of Function in Lesson 1)

Part One: Learning Objectives

Understand the concept of function and the three elements that constitute a function

Can find the domain of some simple functions

Master the concept of the same function and be able to judge

Able to find the function value and range of simple functions

Functions and their representation PPT, part 2: independent learning

Problem guide

Preview the contents of textbook P85-P88 and think about the following questions:

1. What is the concept of function?

2. How are the independent variables and domain of a function defined?

3． How is the range of a function defined?

A preliminary exploration of new knowledge

1. Concepts related to functions

Generally, given two non-empty real number sets A and B, and the corresponding relationship f, if for every real number x in set A, according to the corresponding relationship f, there is a ___________ real number y=f in set B (x) corresponds to x, then f is said to be a function defined on the set A, denoted as ____________________, where x is called __________, y is called __________, and the range of independent variable values ​​(i.e., number set A) This function is called the __________, and the set of all function values ​​is called the range of the function.

■Instructions from famous teachers

5 points on the concept of functions

(1) When A and B are non-empty number sets, the symbol "f: A→B" represents a function from A to B.

(2) The numbers in set A are arbitrary, and the numbers in set B are unique.

(3) The symbol "f" represents the corresponding relationship. The specific meaning of f in different functions is different.

(4) The definition of function emphasizes the "correspondence relationship", which can also be expressed by lowercase English letters such as g and h.

(5) In the representation of functions, it does not matter what letters are used to represent the independent variables and dependent variables. For example, f(x)=2x+1, x∈R and y=2s+1, s∈R are the same function.

2. same function

If two function expressions represent the same function _________ and ___________ is also the same (that is, for each value of the independent variable, the function values ​​obtained by the two function expressions are equal), then the two function expressions are said to represent is the same function.

self-test

Judge whether it is true or false (mark “√” if it is correct and “×” if it is wrong)

(1) A functional relationship can be established between any two sets. ()

(2) A function can be determined if the domain and corresponding relationship are known. ()

(3) According to the definition of the function, each x in the domain of definition can correspond to a different y.()

It is known that the function g(x)=2x2-1, then g(1)=()

A． -1　B． 0

C． 1 D． 2

The domain of function f(x)=14-x is ()

A． (-∞,4) B． (-∞, 4]

C． (4, +∞) D. [4，+∞)

Which of the following formulas cannot represent the function y=f(x) is ()

A． x=y2+1 B． y＝2x2＋1

C． x-2y=6 D. x=y

Functions and their representation PPT, the third part: lecture and practice interaction

The concept of function

(1) As shown in the figure, what can be used as the image of the function y=f(x) is ()

(2) The following three statements:

①If the value domain of the function contains only one element, the domain also contains only one element;

②If f(x)=5(x∈R), then f(π)=5 must be true;

③A function is the correspondence between two sets.

The number of correct statements is ()

A． 0B. 1 C. 2 D. 3

(3) It is known that set A=[0,8] and set B=[0,4], then the following correspondence cannot be regarded as a functional relationship from A to B of is()

A． f：x→y＝18x B． f：x→y＝14x

C． f：x→y＝12x D. f：x→y＝x

regular method

(1) Method to determine whether the given correspondence is a function

① First observe whether the two number sets A and B are non-empty;

② Verify the arbitrariness of x in set A and the uniqueness of y in set B under the corresponding relationship.

(2) Steps to determine whether the corresponding relationship is a function based on the graph

① Take any straight line l perpendicular to the x-axis;

② Move the straight line l parallelly within the definition domain;

③If l has one and only one intersection point with the graph, it is a function; if there is no intersection point or two or more intersection points within the definition domain, it is not a function.

Find the domain of a function

Find the domain of the following functions:

(1)y＝(x＋1)2x＋1-1-x; (2)y＝3-x|x|-5.

regular method

(1) Common methods for finding the domain of a function

①If f(x) is a fraction, you should consider making the denominator non-zero;

②If f(x) is an even-order radical, the radicand is greater than or equal to zero;

③If f(x) is an exponential power, the domain of the function is the set of real numbers that make the power operation meaningful;

④If f(x) is composed of several formulas, then the domain of the function is the intersection of several partial domains;

⑤If f(x) is the analytical formula of the actual problem, it should be consistent with the actual problem to make the actual problem meaningful.

(2) In question (1), it is easy to simplify y = x + 1 - 1 - x, and the domain of the function is found to be {x|x≤1}. When finding the domain of a function, the functional formula cannot be blindly deformed.

same function

(1) Give the following three statements:

①f(x)=x0 and g(x)=1 are the same function; ②y=f(x), x∈R and y=f(x+1), x∈R may be the same function; ③y=f(x ), x∈R is the same function as y=f(t), t∈R.

The number of correct statements is ()

A． 3B． 2

C． 1 D． 0

(2) The following sets of functions:

①f(x)=x2-xx, g(x)=x-1;

②f(x)=xx, g(x)=xx;

③f(x)＝x+1·1－x, g(x)＝1－x2;

④f(x)=(x+3)2, g(x)=x+3.

Among them, the one that represents the same function is ________ (fill in the serial numbers of all the same functions).

Reflection and induction

Three points that should be noted when judging two functions to be the same function

(1) As long as one of the definition domain and the corresponding relationship is different, it is not the same function. Even if the definition domain and value range are the same, they are not necessarily the same function.

(2) A function is the correspondence between two non-empty number sets, so there are no restrictions on the letters used to represent independent variables and dependent variables.

(3) When simplifying the analytical expression, it must be an equivalent deformation.

Find function value and range

It is known that f(x)=12-x(x∈R, x≠2), g(x)=x+4(x∈R).

(1) Find the values ​​of f(1) and g(1);

(2) Find f(g(x)).

Interactive exploration

1. (Various question method) Under the conditions of this example, find the value of g(f(1)) and the expression of f(2x+1).

2. (Variable conditions) If the domain of g(x) in this example is changed to {0, 1, 2, 3}, find the value domain of g(x).

regular method

(1) Method to find function value

① First, determine the specific meaning of the corresponding relationship f of the function;

②Then substitute the variable values ​​into the analytical calculation. For the evaluation of functions of type f(g(x)), proceed in the order of "from inside to outside". Pay attention to f(g(x)) and g(f( The difference between x)).

(2) Common methods for finding the range of functions

① Observation method: For some relatively simple functions, their value ranges can be obtained through observation;

② Matching method: This method is the basic method for finding the value range of "quadratic function class", that is, converting the function through the formula into a method that can directly see its value range;

③Separating constant method: This method is mainly for rational fractions, that is, converting rational fractions into the form of "inverse proportional function class" to facilitate the evaluation domain;

④ Substitution method: that is, using new yuan substitution, the given function is transformed into a function whose value range is easy to determine, so as to obtain the value range of the original function.

Functions and their representation PPT, Part 4: Feedback on achievement of standards

1. If f(x)=x+1, then f(3)=()

A． 2B. 4

C． 22D. 10

2. For function f: A→B, if a∈A, then the following statement is wrong ()

A． f(a)∈B

B． f(a) has and only one

C． If f(a)=f(b), then a=b

D. If a=b, then f(a)=f(b)

3． It is known that function f(x)=2x-3, x∈{x∈N|1≤x≤5}, then the value range of function f(x) is ________.

4. It is known that the function f(x)=6x-1-x+4.

(1) Find the domain of function f(x);

(2) Find the values ​​of f(-1) and f(12).

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Update Time: 2024-09-27

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