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2017_2018学年高中数学第一章集合与函数概念1.3.1.1函数的单调性课时达标训练新人教A版必

1.3.1.1 函数的单调性 课时达标训练 1.设 f(x)=(2a-1)x+b 在 R 上是减函数,则有 A.a≥ B.a≤ C.a>( D.a< ) 【解析】选 D.因为 y=(2a-1)x+b 在 R 上是减函数,所以 2a-1<0,即 a< . 2.下列函数中,定义域为 R 且为增函数的是 ( A.y= C.y=x 2 ) B.y=x 3 D.y=|x| 【解析】选 B.A 中 y= 的定义域为(-∞,0)∪(0,+∞), B 中,y=x 在 R 上为增函数, C 中,y=x 在(0,+∞)上为增函数;在(-∞,0)上为减函数, D 中,y=|x|= 在(0,+∞)上为增函数,在(-∞,0)上为减函数. 2 3 3.已知函数 f(x)的图象如图所示,则函数的单调增区间为________. 【解析】由图知单调增区间为(-∞,-1),(1,+∞). 答案:(-∞,-1),(1,+∞) 4.设函数 f(x)是 R 上的减函数,若 f(m-1)>f(2m-1),则实数 m 的取值范围是________. 【解析】由 f(m-1)>f(2m-1)且 f(x)是 R 上的减函数,得 m-1<2m-1,所以 m>0. 答案:m>0 5.利用单调性的定义证明函数 f(x)= 【证明】设 0<x1<x2,f(x1)-f(x2)= 在(0,+∞)上为单调减函数. = = , 因为 0<x1<x2,所以 >0,x2-x1>0,x1+x2>0, 1 即 >0, 所以 f(x1)-f(x2)>0,即 f(x1)>f(x2), 所以 f(x)在(0,+∞)上为单调减函数. 6.若 f(x)= 是 R 上的单调函数,求实数 a 的取值范围. 【解析】因为 f(x)= 是 R 上的单调函数,所以 解 得:a≥ ,故实数 a 的取值范围为 . 2

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