3.1.2函数的表示法课件-2024-2025学年高一上学期数学人教A版（2019）必修第一册

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PPT模板：www.1ppt.com/moban/ PPT素材：www.1ppt.com/sucai/ PPT背景：www.1ppt.com/beijing/ PPT图表：www.1ppt.com/tubiao/ PPT下载：www.1ppt.com/xiazai/ PPT教程： www.1ppt.com/powerpoint/ 资料下载：www.1ppt.com/ziliao/ 范文下载：www.1ppt.com/fanwen/ 试卷下载：www.1ppt.com/shiti/ 教案下载：www.1ppt.com/jiaoan/ PPT论坛：www.1ppt.cn PPT课件：www.1ppt.com/kejian/ 语文课件：www.1ppt.com/kejian/yuwen/ 数学课件：www.1ppt.com/kejian/shuxue/ 英语课件：www.1ppt.com/kejian/yingyu/ 美术课件：www.1ppt.com/kejian/meishu/ 科学课件：www.1ppt.com/kejian/kexue/ 物理课件：www.1ppt.com/kejian/wuli/ 化学课件：www.1ppt.com/kejian/huaxue/ 生物课件：www.1ppt.com/kejian/shengwu/ 地理课件：www.1ppt.com/kejian/dili/ 历史课件：www.1ppt.com/kejian/lishi/ 5 PPT模板：www.1ppt.com/moban/ PPT素材：www.1ppt.com/sucai/ PPT背景：www.1ppt.com/beijing/ PPT图表：www.1ppt.com/tubiao/ PPT下载：www.1ppt.com/xiazai/ PPT教程： www.1ppt.com/powerpoint/ 资料下载：www.1ppt.com/ziliao/ 范文下载：www.1ppt.com/fanwen/ 试卷下载：www.1ppt.com/shiti/ 教案下载：www.1ppt.com/jiaoan/ PPT论坛：www.1ppt.cn PPT课件：www.1ppt.com/kejian/ 语文课件：www.1ppt.com/kejian/yuwen/ 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美术课件：www.1ppt.com/kejian/meishu/ 科学课件：www.1ppt.com/kejian/kexue/ 物理课件：www.1ppt.com/kejian/wuli/ 化学课件：www.1ppt.com/kejian/huaxue/ 生物课件：www.1ppt.com/kejian/shengwu/ 地理课件：www.1ppt.com/kejian/dili/ 历史课件：www.1ppt.com/kejian/lishi/ 32 PPT模板：www.1ppt.com/moban/ PPT素材：www.1ppt.com/sucai/ PPT背景：www.1ppt.com/beijing/ PPT图表：www.1ppt.com/tubiao/ PPT下载：www.1ppt.com/xiazai/ PPT教程： www.1ppt.com/powerpoint/ 资料下载：www.1ppt.com/ziliao/ 范文下载：www.1ppt.com/fanwen/ 试卷下载：www.1ppt.com/shiti/ 教案下载：www.1ppt.com/jiaoan/ PPT论坛：www.1ppt.cn PPT课件：www.1ppt.com/kejian/ 语文课件：www.1ppt.com/kejian/yuwen/ 数学课件：www.1ppt.com/kejian/shuxue/ 英语课件：www.1ppt.com/kejian/yingyu/ 美术课件：www.1ppt.com/kejian/meishu/ 科学课件：www.1ppt.com/kejian/kexue/ 物理课件：www.1ppt.com/kejian/wuli/ 化学课件：www.1ppt.com/kejian/huaxue/ 生物课件：www.1ppt.com/kejian/shengwu/ 地理课件：www.1ppt.com/kejian/dili/ 历史课件：www.1ppt.com/kejian/lishi/ 33 学 习 目 标 核 心 素 养 1.掌握函数的三种表示方法：解析法、图象法、列表法．(重点) 2．会根据不同的需要选择恰当的方法表示函数．(难点) 1.通过函数表示的图象法培养直观想象素养． 2．通过函数解析式的求法培养运算素养. 函数的表示法 数学表达式 图象 表格 思考：任何一个函数都可以用解析法、列表法、图表法三种形式表示吗？ 提示：不一定． 并不是所有的函数都可以用解析式表示，不仅如此，图象法也不适用于所有函数，如D(x)＝eq \b\lc\{\rc\ (\a\vs4\al\co1(0，x∈Q，,1，x∈∁RQ.))列表法虽在理论上适用于所有函数，但对于自变量有无数个取值的情况，列表法只能表示函数的一个概况或片段． 【例1】　某商场新进了10台彩电，每台售价3 000元，试求售出台数x与收款数y之间的函数关系，分别用列表法、图象法、解析法表示出来． 函数的三种表示方法 [解]　①列表法如下： x(台) 1 2 3 4 5 y(元) 3 000 6 000 9 000 12 000 15 000 x(台) 6 7 8 9 10 y(元) 18 000 21 000 24 000 27 000 30 000 ②图象法：如图所示． ③解析法：y＝3 000x，x∈{1,2,3，…，10}． 列表法、图象法和解析法是从三个不同的角度刻画自变量与函数值的对应关系，同一个函数可以用不同的方法表示．在用三种方法表示函数时要注意：①解析法必须注明函数的定义域；②列表法中选取的自变量要有代表性，应能反映定义域的特征；③图象法中要注意是否连线． 1．(1)某学生离家去学校，一开始跑步前进，跑累了再走余下的路程．下列图中纵轴表示离校的距离，横轴表示出发后的时间，则较符合该学生走法的是(　　) A　 　B　 　C　 　D (2)由下表给出函数y＝f(x)，则f(f(1))等于(　　) x 1 2 3 4 5 y 4 5 3 2 1 A.1　　　　 B．2 C．4 D．5 (1)D　(2)B　[(1)结合题意可知，该生离校的距离先快速减少，又较慢减少，最后到0，故选D. (2)由题意可知，f(1)＝4，f(4)＝2，∴f(f(1))＝f(4)＝2，故选B.] 【例2】　作出下列函数的图象并求出其值域． (1)y＝－x，x∈{0,1，－2,3}；(2)y＝eq \f(2,x)，x∈[2，＋∞)；(3)y＝x2＋2x，x∈[－2,2)． 图象的画法及应用 [解]　(1)列表 x 0 1 －2 3 y 0 －1 2 －3 函数图象只是四个点(0,0)，(1，－1)，(－2,2)，(3，－3)，其值域为{0，－1,2，－3}． (2)列表 x 2 3 4 5 … y 1 eq \f(2,3) eq \f(1,2) eq \f(2,5) … 当x∈[2，＋∞)时，图象是反比例函数y＝eq \f(2,x)的一部分，观察图象可知其值域为(0,1]． (3)列表 x －2 －1 0 1 2 y 0 －1 0 3 8 画图象，图象是抛物线y＝x2＋2x在－2≤x<2之间的部分． 由图可得函数的值域为[－1,8)． 描点法作函数图象的三个关注点 1画函数图象时首先关注函数的定义域，即在定义域内作图. 2图象是实线或实点，定义域外的部分有时可用虚线来衬托整个图象. 3要标出某些关键点，例如图象的顶点、端点、与坐标轴的交点等.要分清这些关键点是实心点还是空心圈. 提醒：函数图象既可以是连续的曲线，也可以是直线、折线、离散的点等. 2．画出下列函数的图象： (1)y＝x＋1(x≤0)； (2)y＝x2－2x(x>1，或x<－1)． [解]　(1)y＝x＋1(x≤0)表示一条射线，图象如图①. (2)y＝x2－2x＝(x－1)2－1(x>1，或x<－1)是抛物线y＝x2－2x去掉－1≤x≤1之间的部分后剩余曲线．如图②. [探究问题] 已知f(x)的解析式，我们可以用代入法求f(g(x))，反之，若已知f(g(x))，如何求f(x)． 提示：若已知f(g(x))的解析式，我们可以用换元法或配凑法求f(x)． 函数解析式的求法 【例3】　(1)已知f(eq \r(x)＋1)＝x－2eq \r(x)，则f(x)＝________； (2)已知函数f(x)是一次函数，若f(f(x))＝4x＋8，则f(x)＝________； (3)已知函数f(x)对于任意的x都有f(x)－2f(－x)＝1＋2x，则f(x)＝________. [思路点拨]　(1)用换元法或配凑法求解；(2)用待定系数法求解；(3)用方程组法求解． (1)x2－4x＋3(x≥1)　(2)2x＋eq \f(8,3)或－2x－8　(3)eq \f(2,3)x－1　[(1)法一(换元法)：令t＝eq \r(x)＋1，则t≥1，x＝(t－1)2，代入原式有f(t)＝(t－1)2－2(t－1)＝t2－4t＋3，f(x)＝x2－4x＋3(x≥1)． 法二(配凑法)：f(eq \r(x)＋1)＝x＋2eq \r(x)＋1－4eq \r(x)－4＋3＝(eq \r(x)＋1)2－4(eq \r(x)＋1)＋3， 因为eq \r(x)＋1≥1， 所以f(x)＝x2－4x＋3(x≥1)． (2)设f(x)＝ax＋b(a≠0)， 则f(f(x))＝f(ax＋b)＝a(ax＋b)＋b＝a2x＋ab＋b. 又f(f(x))＝4x＋8， 所以a2x＋ab＋b＝4x＋8， 即eq \b\lc\{\rc\ (\a\vs4\al\co1(a2＝4，,ab＋b＝8，))解得eq \b\lc\{\rc\ (\a\vs4\al\co1(a＝2，,b＝\f(8,3)))或eq \b\lc\{\rc\ (\a\vs4\al\co1(a＝－2，,b＝－8.)) 所以f(x)＝2x＋eq \f(8,3)或f(x)＝－2x－8. (3)由题意，在f(x)－2f(－x)＝1＋2x中，以－x代x可得f(－x)－2f(x)＝1－2x，联立可得eq \b\lc\{\rc\ (\a\vs4\al\co1(fx－2f－x＝1＋2x，,f－x－2fx＝1－2x，))消去f(－x)可得f(x)＝eq \f(2,3)x－1.] 1．(变条件)把本例(2)的题干改为“已知函数f(x)是二次函数，且f(0)＝1，f(x＋1)－f(x)＝2x.”求f(x)的解析式． [解]　设f(x)＝ax2＋bx＋c，由f(0)＝1得c＝1. 又f(x＋1)＝a(x＋1)2＋b(x＋1)＋1， ∴f(x＋1)－f(x)＝2ax＋a＋b. 由2ax＋a＋b＝2x，得eq \b\lc\{\rc\ (\a\vs4\al\co1(2a＝2，,a＋b＝0，)) 解得a＝1，b＝－1. ∴f(x)＝x2－x＋1. 2．(变条件)把本例(3)的题干改为“2feq \b\lc\(\rc\)(\a\vs4\al\co1(\f(1,x)))＋f(x)＝x(x≠0)”，求f(x)的解析式． [解]　f(x)＋2feq \b\lc\(\rc\)(\a\vs4\al\co1(\f(1,x)))＝x，令x＝eq \f(1,x)， 得feq \b\lc\(\rc\)(\a\vs4\al\co1(\f(1,x)))＋2f(x)＝eq \f(1,x). 于是得关于f(x)与feq \b\lc\(\rc\)(\a\vs4\al\co1(\f(1,x)))的方程组 eq \b\lc\{\rc\ (\a\vs4\al\co1(fx＋2f\b\lc\(\rc\)(\a\vs4\al\co1(\f(1,x)))＝x，,f\b\lc\(\rc\)(\a\vs4\al\co1(\f(1,x)))＋2fx＝\f(1,x).)) 解得f(x)＝eq \f(2,3x)－eq \f(x,3)(x≠0)． 求函数解析式的四种常用方法 1待定系数法：若已知fx的解析式的类型，设出它的一般形式，根据特殊值确定相关的系数即可. 2换元法：设t＝gx，解出x，代入fgx，求ft的解析式即可. 3配凑法：对fgx的解析式进行配凑变形，使它能用gx表示出来，再用x代替两边所有的“gx”即可. 4方程组法或消元法：当同一个对应关系中的两个之间有互为相反数或互为倒数关系时，可构造方程组求解. 提醒：应用换元法求函数解析式时，务必保证函数在换元前后的等价性. 1．函数有三种常用的表示方法，可以适时的选择，以最佳的方式表示函数． 2．作函数图象必须要让作出的图象反映出图象的伸展方向，与x轴、y轴有无交点，图象有无对称性，并标明特殊点． 3．求函数解析式的主要方法有：代入法、待定系数法、换元法、解方程组法(消元法)，注意有的函数要注明定义域. 1．思考辨析 (1)任何一个函数都可以用解析法表示．(　　) (2)函数的图象一定是定义区间上一条连续不断的曲线．(　　) [答案]　(1)×　(2)× 2．已知函数f(x＋1)＝3x＋2，则f(x)的解析式是(　　) A．f(x)＝3x－1　　 B．f(x)＝3x＋1 C．f(x)＝3x＋2 D．f(x)＝3x＋4 A　[令x＋1＝t，则x＝t－1，∴f(t)＝3(t－1)＋2＝3t－1.∴f(x)＝3x－1.] 3．已知函数f(x)，g(x)分别由下表给出． 则g(f(5))＝________；f(g(2))＝________. 4　3　[由题表可知f(5)＝3，g(3)＝4，∴g(f(5))＝g(3)＝4. 又g(2)＝5，f(5)＝3， ∴f(g(2))＝f(5)＝3.] 4．已知函数f(x)＝x2－2x(－1≤x≤2)． (1)画出f(x)图象的简图； (2)根据图象写出f(x)的值域． [解]　(1)f(x)图象的简图如图所示． (2)观察f(x)的图象可知，f(x)图象上所有点的纵坐标的取值范围是[－1,3]， 即f(x)的值域是[－1,3]． $$