"Piecewise Function" Concept and Properties of Functions PPT

"Piecewise Function" Concept and Properties of Functions PPT

Part One: Explanation of Curriculum Standards

1. Understand the concept of piecewise functions.

2. Be able to find the function value of piecewise functions and draw the graph of piecewise functions.

3. Be able to list piecewise functions in actual problems and solve related problems.

Piecewise function PPT, part 2: independent preview

piecewise function

1. (1) Textbook P68 Example 5, when drawing the function graph, simplify the function y=|x| to get

y={■(x"," x≥0"," @"-" x"," x<0"." )┤What are the characteristics of this function?

Tip: When x≥0 and x<0, the function expression is different, that is, the corresponding relationship is different.

(2) Draw the graph of the function y=2x(x∈R), and then draw the graph of y=x2(x∈R). Putting these two graphs in the same rectangular coordinate system can also represent the function graph Like?

Tip: The functions y=2x(x∈R) and y=x2(x∈R) together cannot represent the function graph, because when taking a certain x value, the y value is not necessarily unique.

(3) Draw the graphs of functions y=2x (x<0) and y=x2 (x≥0) respectively in the same rectangular coordinate system. Can these two function graphs together represent the function graph? ?How to write its analytical expression?

Tip: The function graph can be expressed because it conforms to the function definition and the analytical expression can be written as

y={■(2x"," x<0"," @x^2 "," x≥0"." )┤

(4) Similar to y={■(x"," x≥0"," @"-" x"," x<0)┤ and y={■(2x"," x<0"," @x ^2 "," The function of x≥0)┤ is called a piecewise function. Is the piecewise function one function or two functions?

Tip: No matter how many pieces a piecewise function is divided into, it is still one function. Do not mistake it for several functions.

(5) Please give some examples of piecewise functions in real life.

Tip: In real life, taxi billing, telecommunications rates, personal income tax, etc. are all piecewise functions.

2. Fill in the blanks

If a function y=f(x), x∈A has different corresponding relationships according to the different value ranges of the independent variable x in A, then such a function is called a piecewise function.

Piecewise function PPT, the third part: inquiry learning

Evaluate piecewise functions

Example 1 Known function f(x)={■(x+2"," x<0"," @x^2 "," 0≤x<2"," @1/2 x"," x≥ 2"," )┤

(1) Find the value of f(f(f("-" 1/2)));

(2) If f(x)=2, find the value of x.

(2) Let x+2=2, x2=2, 1/2x=2 respectively, find x piece by piece and verify.

Reflection and insights 1. Steps to find the function value of piecewise function

(1) First determine which interval the independent variable corresponding to the evaluated value belongs to.

(2) Then substitute the analytical expression corresponding to the paragraph to evaluate until the value is obtained. When the form of f(f(x0)) appears, the evaluation should be performed from the inside to the outside.

2. Steps to find the value of the independent variable if the function value is known

(1) First determine the independent variables, possible intervals and their corresponding functional analytical formulas.

(2) Then substitute the function values ​​into different analytical expressions.

(3) Find the value of the independent variable by solving the equation.

(4) Check whether the value sought is within the interval in question.

Piecewise function PPT, part 4: thinking methods

Use the combination of numbers and shapes to find the number of roots of an equation

Typical example: For different value ranges of m, discuss the number of real roots of the equation x2-4|x|+5=m.

Analysis: You can consider the graph of the function corresponding to the left side of the given equation, that is, draw the graph of the function y=x2-4|x|+5, and see the number of intersections between the graph and the straight line y=m. Changes lead to conclusions.

Solution: Convert the problem of the number of real roots of the equation x2-4|x|+5=m into the image of the function y=x2-4|x|+5 and the straight line y=m The problem of the number of intersection points.

y=x2-4|x|+5={■(x^2 "-" 4x+5"," x≥0"," @x^2+4x+5"," x<0 "," )┤

Make an image as shown in the figure.

When m<1, the straight line y=m has no intersection with the image, so the equation has no solution.

When m=1, the straight line y=m has two intersection points with the image,

Therefore the equation has two real roots.

When 1

Therefore the equation has four real roots.

When m=5, the straight line y=m has three intersection points with the image,

Therefore the equation has three real roots.

When m>5, the straight line y=m has two intersection points with the image,

Therefore the equation has two real roots.

Reflection on this question: Through constructing functions and using the idea of ​​​​combination of numbers and shapes, the number of real roots can be intuitively and vividly obtained through images. However, it should be noted that this method generally only finds the number of roots, and does not need to know the specific values ​​of the real roots. .

Piecewise function PPT, part 5: practice in class

1. It is known that f(x)={■(x^2 "," x>0"," @π"," x=0"," @0"," x<0"," )┤then f (f(-3)) is equal to ()

A.0B.πC.π2D.9

Analysis: f(f(-3))=f(0)=π.

Answer:B

2. The image of function f(x)=x+("|" x"|" )/x is ()

Analysis: f(x)=x+("|" x"|" )/x={■(x+1"," x>0"," @x"-" 1"," x<0)┤ is a piecewise function.

Answer:C

3. The method used by a passenger transport company to determine the ticket price is: if the journey does not exceed 100 kilometers, the fare is 0.5 yuan per kilometer. If it exceeds 100 kilometers, the excess part is priced at 0.4 yuan per kilometer. The passenger fare y The functional relationship between (yuan) and the number of kilometers traveled x (kilometers) is _____.

Analysis: Find the analytical formula based on whether the trip is greater than 100 kilometers.

Answer: y={■(0"." 5x"," 0≤x≤100"," @10+0"." 4x"," x>100)┤

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