"Parity" Concept and Properties of Functions PPT

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"Parity" Concept and Properties of Functions PPT

Part One: Explanation of Curriculum Standards

1. Understand the definitions of odd functions and even functions based on specific functions.

2. Understand the characteristics of the graphs of odd functions and even functions.

3. Be able to judge (or prove) the parity of functions.

Parity PPT, part 2: independent preview

1. Even functions

1. (1) Observe the graphs of the following functions. Can you summarize the common characteristics of these three functions through the graphs of these functions?

Tip: The domains of these three functions are symmetrical about the origin, and the images are symmetrical about the y-axis.

(2) For the above three functions, what is the relationship between f(1) and f(-1), f(2) and f(-2), f(3) and f(-3)? This shows that about the y-axis What is the relationship between the coordinates of symmetrical points?

Tip: f(1)=f(-1), f(2)=f(-2), f(3)=f(-3). The abscissas of points that are symmetrical about the y-axis are opposite numbers to each other, and the vertical coordinates are opposite numbers to each other. The coordinates are equal.

(3) Generally speaking, if the graph of function y=f(x) is symmetrical about the y-axis, what is the relationship between f(x) and f(-x)? Is the opposite true?

Tip: If the graph of the function y=f(x) is symmetrical about the y-axis, then f(x)=f(-x). On the contrary, if f(x)=f(-x), then the function y=f( The graph of x) is symmetrical about the y-axis.

2. Fill in the blanks

(1) Definition: Generally speaking, let the domain of function f(x) be I. If ∀x∈I, both have -x∈I, and f(-x)=f(x), then the function f(x ) is called an even function.

(2) Image characteristics of even functions: The image is symmetrical about the y-axis.

3. Do it:

Among the following functions, which one is an even function ()

A.f(x)=x2

B.f(x)=x

C.f(x)=

D.f(x)=x+x3

Answer:A

2. Odd functions

1. (1) Observe the graphs of functions f(x)=x and f(x)= (as shown in the figure). Can you find any common characteristics of the graphs of these two functions?

Tip: It is easy to get that the domain is symmetrical about the origin and the image is symmetrical about the origin.

(2) What is the relationship between the above two functions f(1) and f(-1), f(2) and f(-2), f(3) and f(-3)?

Hint: f(-1)=-f(1), f(-2)=-f(2), f(-3)=-f(3).

(3) Generally speaking, if the graph of function y=f(x) is symmetrical about the origin, what is the relationship between f(x) and f(-x)? Is the opposite true?

Tip: If the graph of function y=f(x) is symmetrical about the origin, then f(-x)=-f(x). On the contrary, if f(-x)=-f(x), then function y=f The graph of (x) is symmetrical about the origin.

2. Similar to the definition of even function, try to fill in the blanks

(1) Definition: Generally speaking, let the domain of function f(x) be I. If ∀x∈I, both have -x∈I, and f(-x)=-f(x), then the function f( x) is called an odd function.

(2) Image characteristics of odd functions: the image is symmetrical about the origin.

3. Do it

(1) The graph of function f(x)= -x is symmetric about ().

A.y axis B.straight line y=-x

C. Coordinate origin D. Line y=x

(2) The function represented by the following graph has parity and evenness ()

Analysis: (1) Because f(x) = -x is an odd function, the graph of this function is symmetrical about the origin of the coordinates.

(2) The graph of the function in option A is not symmetrical about the origin or the y-axis, so it is excluded; the domain of the function represented by the graph in options C and D is not symmetrical about the origin and does not have parity, so it is excluded; option The graph in B is symmetric about the y-axis, and the function it represents is an even function. Therefore, choose B.

Answer:(1)C　(2)B

Parity PPT, Part 3: Inquiry Learning

Determine the parity of a function

Example 1 Determine the parity of the following functions:

(1)f(x)=(2x^2+2x)/(x+1);

(2)f(x)=x3-2x;

(3)f(x)=√(1"-" x^2 )+√(x^2 "-" 1);

(4)f(x)={■(x"(" 1"-" x")," x<0"," @x"(" 1+x")," x>0"." )┤

Analysis: When using the definitions of odd and even functions to determine the parity of a function, first find the domain of the function to see if it is symmetrical about the origin. If the domain is symmetrical about the origin, then determine f(-x) and f(x ) relationship. In order to determine the relationship between f(-x) and f(x), you can either start with f(-x) to simplify and organize, or you can consider f(-x)+f(x) or f(-x )-f(x) is equal to 0. When f(x) is not equal to 0, you can also consider the relationship between (f"(-" x")" )/(f"(" x")" ) and 1 or -1 relationship, you can also consider using the image method.

Solution: (1) The domain of the function is {x|x≠-1}, which is not symmetrical about the origin, so f(x) is neither an odd function nor an even function.

(2) The domain of the function is R, symmetrical about the origin, f(-x)=(-x)3-2(-x)=2x-x3=-f(x), ∴f(x) is an odd function .

Parity PPT, part 4: thinking methods

Use definition method and assignment method to solve the problem of parity of abstract functions

For example, if the function f(x) defined on R satisfies: for any x1, x2∈R, f(x1+x2)=f(x1)+f(x2), and when x>0, f(x)<0, then ()

A.f(x) is an odd function and an increasing function on R

B.f(x) is an odd function and a decreasing function on R

C.f(x) is an odd function and is not a monotonic function on R

D. The monotonicity and parity of f(x) cannot be determined

Analysis: Let x1=x2=0, then f(0)=2f(0),

So f(0)=0.

Let x1=x,x2=-x,

Then f(-x)+f(x)=f(x-x)=f(0)=0,

So f(-x)=-f(x), so the function y=f(x) is an odd function.

Assume x1

Since x2-x1>0, so f(x2-x1)<0,

Therefore f(x2)

Therefore, the function y=f(x) is a decreasing function on R. Therefore, choose B.

Answer:B

Reflection and Insight 1. To judge the parity of an abstract function, you should use the definition of parity of a function to find the right direction, clever assignment, reasonable and flexible deformation, and find out the relationship between f(-x) and f(x), so as to judge or prove Parity of abstract functions.

2. Sometimes it is necessary to study the sum of f(-x)+f(x) as a whole.

For example: In the above typical example, using f(-x)+f(x)=0, we can conclude that y=f(x) is an odd function.

Parity PPT, Part 5: Practice in class

1. It is known that the domain of an odd function is {-1,2,a,b}, then a+b is equal to ()

A.-1 B.1 C.0 D.2

Analysis: Because the domain of an odd function is {-1,2,a,b},

According to the definition domain of the odd function is symmetric about the origin,

So one of a and b is equal to 1, and the other is equal to -2,

So a+b=1+(-2)=-1.

Answer:A

2. Function y=(x^2 "(" x+4")" )/(x+4)()

A. It is an odd function B. It is an even function

C. It is both an odd function and an even function D. It is neither an odd function nor an even function

Analysis: From the question, we know that the domain of the function is (-∞,-4)∪(-4,+∞), which is not symmetrical about the origin, so the function is neither an odd function nor an even function.

Answer:D

3. Let f(x) be an odd function defined on R. When x≤0, f(x)=2x2-x, then f(1)=()

A.-1 B.-3 C.1 D.3

Analysis: When x≤0, f(x)=2x2-x, f(-1)=2×(-1)2-(-1)=3. Because f(x) is an odd value defined on R function,

Therefore, f(1)=-f(-1)=-3, so choose B.

Answer:B

4. If the function f(x)=(x+a)(x-4) is an even function, then the real number a=________.

Analysis: f(x)=x2+(a-4)x-4a,

∵f(x) is an even function, ∴a-4=0, that is, a=4.

Answer: 4

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Update Time: 2024-07-31

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