《分段函數(shù)》函數(shù)的概念與性質(zhì)PPT
第一部分內(nèi)容：課標(biāo)闡釋
1.了解分段函數(shù)的概念.
2.會求分段函數(shù)的函數(shù)值,能畫出分段函數(shù)的圖象.
3.能在實際問題中列出分段函數(shù),并能解決有關(guān)問題.
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分段函數(shù)PPT，第二部分內(nèi)容：自主預(yù)習(xí)
分段函數(shù)
1.(1)教材P68例5,在畫函數(shù)圖象時,將函數(shù)y=|x|化簡得到
y={■(x"," x≥0"," @"-" x"," x<0"." )┤這個函數(shù)有什么特點?
提示:當(dāng)x≥0和x<0時,這個函數(shù)表達(dá)式不一樣,也就是對應(yīng)關(guān)系不同.
(2)作出函數(shù)y=2x(x∈R)的圖象,再作出y=x2(x∈R)的圖象.把這兩個圖象放在同一個直角坐標(biāo)系中還能表示函數(shù)圖象嗎?
提示:函數(shù)y=2x(x∈R)和y=x2(x∈R)合起來不能表示函數(shù)圖象,因為取某個x值時,y值不一定唯一.
(3)在同一個直角坐標(biāo)系中分別畫出函數(shù)y=2x(x<0)和y=x2(x≥0)的圖象,這兩個函數(shù)圖象合起來還能表示函數(shù)圖象嗎?如何寫它的解析式?
提示:可以表示函數(shù)圖象,因為符合函數(shù)定義,解析式可寫為
y={■(2x"," x<0"," @x^2 "," x≥0"." )┤
(4)類似y={■(x"," x≥0"," @"-" x"," x<0)┤和y={■(2x"," x<0"," @x^2 "," x≥0)┤的函數(shù)叫分段函數(shù).分段函數(shù)是一個函數(shù)還是兩個函數(shù)?
提示:不管分段函數(shù)分了幾段,它都是一個函數(shù),不要把它誤認(rèn)為是幾個函數(shù).
(5)請舉出幾個實際生活中分段函數(shù)的例子.
提示:實際生活中,出租車的計費、電信資費、個人所得稅額等均是分段函數(shù).
2.填空
如果函數(shù)y=f(x),x∈A,根據(jù)自變量x在A中不同的取值范圍,有著不同的對應(yīng)關(guān)系,則稱這樣的函數(shù)為分段函數(shù).
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分段函數(shù)PPT，第三部分內(nèi)容：探究學(xué)習(xí)
求分段函數(shù)的求值
例1已知函數(shù)f(x)={■(x+2"," x<0"," @x^2 "," 0≤x<2"," @1/2 x"," x≥2"," )┤
(1)求f(f(f("-" 1/2)))的值;
(2)若f(x)=2,求x的值.
(2)分別令x+2=2,x2=2,1/2x=2,分段求x并驗證.
反思感悟 1.求分段函數(shù)的函數(shù)值的步驟
(1)先確定所求值對應(yīng)的自變量屬于哪一段區(qū)間.
(2)再代入該段對應(yīng)的解析式進(jìn)行求值,直到求出值為止.當(dāng)出現(xiàn)f(f(x0))的形式時,應(yīng)從內(nèi)到外依次求值.
2.已知函數(shù)值求自變量取值的步驟
(1)先確定自變量,可能存在的區(qū)間及其對應(yīng)的函數(shù)解析式.
(2)再將函數(shù)值代入到不同的解析式中.
(3)通過解方程求出自變量的值.
(4)檢驗所求的值是否在所討論的區(qū)間內(nèi).
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分段函數(shù)PPT，第四部分內(nèi)容：思想方法
利用數(shù)形結(jié)合思想求方程根的個數(shù)
典例 對于m不同的取值范圍,討論方程x2-4|x|+5=m的實根的個數(shù).
分析:可考慮給定方程左側(cè)對應(yīng)函數(shù)的圖象,即畫出函數(shù)y=x2-4|x|+5的圖象,看圖象與直線y=m的交點個數(shù)的變化便可得出結(jié)論.
解:將方程x2-4|x|+5=m實根的個數(shù)問題轉(zhuǎn)化為函數(shù)y=x2-4|x|+5的圖象與直線y=m的交點個數(shù)問題.
y=x2-4|x|+5={■(x^2 "-" 4x+5"," x≥0"," @x^2+4x+5"," x<0"," )┤
作出圖象,如圖所示.
當(dāng)m<1時,直線y=m與該圖象無交點,故方程無解.
當(dāng)m=1時,直線y=m與該圖象有兩個交點,
故方程有兩個實根.
當(dāng)1<m<5時,直線y=m與該圖象有四個交點,
故方程有四個實根.
當(dāng)m=5時,直線y=m與該圖象有三個交點,
故方程有三個實根.
當(dāng)m>5時,直線y=m與該圖象有兩個交點,
故方程有兩個實根.
反思感悟 本題通過構(gòu)造函數(shù),利用數(shù)形結(jié)合的思想,直觀形象地通過圖象得出實數(shù)根的個數(shù).但要注意這種方法一般只求根的個數(shù),不需知道實數(shù)根的具體數(shù)值.
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分段函數(shù)PPT，第五部分內(nèi)容：隨堂演練
1.已知f(x)={■(x^2 "," x>0"," @π"," x=0"," @0"," x<0"," )┤則f(f(-3))等于( )
A.0B.πC.π2D.9
解析:f(f(-3))=f(0)=π.
答案:B
2.函數(shù)f(x)=x+("|" x"|" )/x的圖象是( )
解析:f(x)=x+("|" x"|" )/x={■(x+1"," x>0"," @x"-" 1"," x<0)┤是分段函數(shù).
答案:C
3.某客運公司確定客票價格的方法是:如果行程不超過100千米,票價是每千米0.5元,如果超過100千米,超過部分按每千米0.4元定價,則客運票價y(元)與行程千米數(shù)x(千米)之間的函數(shù)關(guān)系式是_____.
解析:根據(jù)行程是否大于100千米來求出解析式.
答案:y={■(0"." 5x"," 0≤x≤100"," @10+0"." 4x"," x>100)┤
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