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函数知识的核心内容 1、函数的单调性 2、函数的奇偶性 3、函数的图象与性质的综合运用 单调性的应用(局部特征) 当x1<x2时 都有f(x1)<f(x2) ⑴ ⑵ 函数f(x)在区间D上是增函数 ⑶ 当x1<x2时 都有f(x1)>f(x2) 函数f(x)在区间D上是减函数 题型1:由(1)(2)推出(3) 应用:单调性的证明 题型2:由(2)(3)推出(1) 应用:求自变量的取值范围 题型3:由(1)(3)推出(2) 应用:可得因变量的大小 奇偶性的应用(整体特征) 对比 代数性质 几何性质 奇函数 图象关于原点对称 f (? x ) ? ? f ( x ) 偶函数 f (? x ) ? f ( x ) 图象关于y轴对称 例题1、函数 f ( x) ? 4 x ? kx ? 5 ,当x ? ( ?2,??) 时是增函数, f (1) _________。 x ? ( ?? , ?2 ) 当 时是减函数,则 的值为 2 变式1、函数 f ( x) ? 4 x 2 ? kx ? 8 在[5,20]上为单调函 数,求实数 k 的取值范围。 1 变式2、函数 f ( x) ? ax ? (a ? 1) x ? 5 ,在 ( ,1) 上为单 2 调增函数,求实数 a 的取值范围。 2 1 例题2、证明:函数 f ( x) ? x ? 在 ?? 1,0? 上为增函数。 x 题型:由(1)(2)推出(3),运用定义 1 变式1、讨论函数 f ( x) ? x ? 的单调性。 x x ?1 变式2、讨论函数 f ( x ) ? lg 的单调性、最小值 x 2 例题 3 、已知y ? f ( x) 是定义在 (?1,1) 上的减函数, 且 f (1 ? a) ? f (2a ?1) ,则 a 的取值范围是______ 变式 1 、已知y ? f ( x) 是定义在 (?1,1) 上的奇函数, [0,1) 函数在 上单调递增,满足 , 2 f ( 1 ? m ) ? f ( m ? 1) ? 0 ______ 则实数 的取值范围是 m 变式2、 f(x)是R上的偶函数,在(0,? ?) 上是减函数, 则p=f(-0.75)与q=f(a2-a+1)(a ? R) 的大小关系是 奇偶性的应用 2 2 f ( x ) ? ( m ? 1 ) x ? (n ? 2) x ? (m ? 1) 例题1、已知函数 问: (1)当 m , n 为何值时,为奇函数? (2)当 m , n 为何值时,为偶函数? 变式1、已知函数 是定义在 a ? 1,2a 上的偶函数,求函数 f ( x )的值域。 ? ? f ( x) ? ax2 ? bx ? 3a ? b 奇偶性的应用 例题1、已知函数 f ( x) ? x ? ax ? bx ? 8 5 3 且 f (?2) ? 10 ,则 f ( 2) ? _____ 变式1、已知函数 f ( x ), g( x ) 都为R 上奇函数 且F ( x ) ? af ( x ) ? bg( x ) ,F ( 2) ? 5 则 F ( ?2) ? _____ 单调性、奇偶性的综合应用 1 1 ? ) x. 1、已知 f ( x) ? ( x 2 ?1 2 ) (1)函数 f ( x的定义域; (2)判断函数的奇偶性; (3)求证. f ( x) ? 0 2.已知函数f(x)是定义为(0,+ ?)上的增函数, 且满足f(xy)=f(x)+f(y),(x,y ? R+),f(2)=1 求 1) f(1); 2)满足f(x)+f(x-3) ? 1的x的取值范围 3) 满足f(x)+f(x-3) < 2的x的取值范围

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