已知幂函数 f(x)=x 的图象经过点 (2, 2 ) ，则 f(9)= （） 3 9 81 1 3 A 幂函数性质幂函数待定系数指数运算新课标II卷 2024

A 代入已知点： 2 α = 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> = 2 2 1 。由指数函数的单调性（同底比较指数）得 α = 2 1 ，即 f ( x ) = x <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> 。故 f ( 9 ) = 9 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> = 3 ，选 A。

高考数学考点题库/幂函数与函数模型/幂函数性质

2024 年新课标II卷·第4题

单选题★★☆☆☆ 常规改编题通法

已知幂函数

f (x) = x^{α}

的图象经过点

(2, 2)

，则

f (9) =

（　　）

3

9

81

\frac{1}{3}

答案A

分步解析

1代入已知点： $2^{α} = 2 = 2^{\frac{1}{2}}$ 。
2由指数函数的单调性（同底比较指数）得 $α = \frac{1}{2}$ ，即 $f (x) = x$ 。
3故 $f (9) = 9 = 3$ ，选 A。

点评幂函数只有一个待定参数

α

，一个点即可确定；注意它与指数函数

y = a^{x}

的区别：底数与指数谁是变量。

#幂函数#待定系数#指数运算

2024 年真题新课标II卷

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