2025年抽象函數(shù)高考試題及答案

一、單項選擇題1.已知函數(shù)\(f(x)\)對于任意實數(shù)\(x\)，\(y\)都滿足\(f(x+y)=f(x)+f(y)\)，且\(f(2)=4\)，則\(f(1)\)等于（）A.\(-2\)B.\(0\)C.\(1\)D.\(2\)答案：D2.設函數(shù)\(f(x)\)是定義在\(R\)上的奇函數(shù)，當\(x\gt0\)時，\(f(x)=x^2-2x\)，則\(f(-1)\)的值為（）A.\(-1\)B.\(1\)C.\(3\)D.\(-3\)答案：A3.已知函數(shù)\(f(x)\)滿足\(f(x+3)=f(x)\)，且當\(x\in[0,3)\)時，\(f(x)=2^x\)，則\(f(2025)\)的值為（）A.\(1\)B.\(2\)C.\(4\)D.\(8\)答案：A4.若函數(shù)\(f(x)\)是定義在\(R\)上的偶函數(shù)，且在\((-\infty,0]\)上單調遞減，則（）A.\(f(-2)\ltf(1)\ltf(3)\)B.\(f(1)\ltf(-2)\ltf(3)\)C.\(f(3)\ltf(-2)\ltf(1)\)D.\(f(3)\ltf(1)\ltf(-2)\)答案：B5.已知函數(shù)\(f(x)\)對于任意\(x_1\)，\(x_2\inR\)，當\(x_1\ltx_2\)時，都有\(zhòng)(f(x_1)\ltf(x_2)\)，則函數(shù)\(f(x)\)是（）A.偶函數(shù)且在\(R\)上單調遞增B.偶函數(shù)且在\(R\)上單調遞減C.奇函數(shù)且在\(R\)上單調遞增D.奇函數(shù)且在\(R\)上單調遞減答案：C6.函數(shù)\(f(x)\)滿足\(f(1-x)=f(1+x)\)，則函數(shù)\(f(x)\)的圖象關于（）對稱A.直線\(x=0\)B.直線\(x=1\)C.點\((1,0)\)D.點\((0,0)\)答案：B7.已知函數(shù)\(f(x)\)是定義在\(R\)上的函數(shù)，且\(f(x+2)=-f(x)\)，當\(x\in[0,2]\)時，\(f(x)=x\)，則\(f(5)\)的值為（）A.\(-1\)B.\(0\)C.\(1\)D.\(2\)答案：C8.若函數(shù)\(f(x)\)滿足\(f(x)=f(-x)\)且\(f(x+2)=f(x)\)，則函數(shù)\(f(x)\)是（）A.周期為\(2\)的奇函數(shù)B.周期為\(2\)的偶函數(shù)C.周期為\(4\)的奇函數(shù)D.周期為\(4\)的偶函數(shù)答案：B9.函數(shù)\(f(x)\)對于任意\(x\inR\)都有\(zhòng)(f(x+4)=f(x)\)，且\(f(x)\)在\([-2,2]\)上的圖象是一段拋物線，已知\(f(-2)=0\)，\(f(0)=-2\)，\(f(2)=0\)，則\(f(7)\)的值為（）A.\(-2\)B.\(0\)C.\(2\)D.\(4\)答案：A10.已知函數(shù)\(f(x)\)滿足\(f(x+1)=\frac{1}{f(x)}\)，當\(x\in(0,1]\)時，\(f(x)=2^x\)，則\(f(\frac{5}{2})\)的值為（）A.\(\frac{1}{2}\)B.\(1\)C.\(\sqrt{2}\)D.\(2\)答案：A二、多項選擇題1.已知函數(shù)\(f(x)\)滿足\(f(x+y)=f(x)+f(y)\)，則下列說法正確的是（）A.\(f(0)=0\)B.\(f(x)\)是奇函數(shù)C.若\(f(1)=2\)，則\(f(2)=4\)D.\(f(x)\)在\(R\)上一定是單調函數(shù)答案：ABC2.設函數(shù)\(f(x)\)是定義在\(R\)上的偶函數(shù)，且在\((0,+\infty)\)上單調遞減，則（）A.\(f(-3)=f(3)\)B.\(f(-2)\gtf(1)\)C.\(f(x)\)在\((-\infty,0)\)上單調遞增D.若\(f(a)=f(b)\)，則\(a=b\)答案：ABC3.已知函數(shù)\(f(x)\)對于任意\(x\inR\)都有\(zhòng)(f(x+2)=f(x)\)，則下列說法正確的是（）A.\(f(x)\)的周期為\(2\)B.\(f(x)\)的圖象關于直線\(x=1\)對稱C.\(f(1)=f(3)\)D.\(f(x)\)在\([0,2]\)上的圖象與在\([2,4]\)上的圖象相同答案：ACD4.若函數(shù)\(f(x)\)滿足\(f(x+1)=-f(x)\)，則（）A.\(f(x)\)的周期為\(2\)B.\(f(x)\)是奇函數(shù)C.\(f(0)=f(2)\)D.\(f(x)\)在\([0,1]\)上與在\([1,2]\)上的函數(shù)值互為相反數(shù)答案：ACD5.已知函數(shù)\(f(x)\)是定義在\(R\)上的函數(shù)，且\(f(x)\)滿足\(f(1-x)=f(1+x)\)，則下列說法正確的是（）A.函數(shù)\(f(x)\)的圖象關于直線\(x=1\)對稱B.\(f(0)=f(2)\)C.\(f(-1)=f(3)\)D.若\(f(x)\)在\((1,+\infty)\)上單調遞增，則在\((-\infty,1)\)上單調遞減答案：ABC6.函數(shù)\(f(x)\)滿足\(f(x+4)=f(x)\)，且\(f(x)\)在\([-2,2]\)上是奇函數(shù)，當\(x\in[0,2]\)時，\(f(x)=x^2\)，則（）A.\(f(-1)=-1\)B.\(f(5)=1\)C.\(f(7)=-1\)D.\(f(-3)=1\)答案：ABC7.已知函數(shù)\(f(x)\)對于任意\(x_1\)，\(x_2\inR\)，當\(x_1\neqx_2\)時，都有\(zhòng)(\frac{f(x_1)-f(x_2)}{x_1-x_2}\gt0\)，則（）A.\(f(x)\)在\(R\)上單調遞增B.若\(x_1\ltx_2\)，則\(f(x_1)\ltf(x_2)\)C.\(f(x)\)是偶函數(shù)D.\(f(x)\)的圖象可能關于點\((1,0)\)對稱答案：ABD8.若函數(shù)\(f(x)\)滿足\(f(x)=f(4-x)\)，且\(f(x)\)在\((2,+\infty)\)上單調遞減，則（）A.函數(shù)\(f(x)\)的圖象關于直線\(x=2\)對稱B.\(f(1)\gtf(3)\)C.\(f(x)\)在\((-\infty,2)\)上單調遞增D.若\(f(a)=f(b)\)，則\(a+b=4\)答案：ABD9.已知函數(shù)\(f(x)\)滿足\(f(x+3)=\frac{1}{f(x)}\)，則（）A.\(f(x)\)的周期為\(6\)B.\(f(1)=f(7)\)C.\(f(2)=f(8)\)D.\(f(x)\)在一個周期內的函數(shù)值不可能相同答案：ABC10.函數(shù)\(f(x)\)對于任意\(x\inR\)都有\(zhòng)(f(x+2)=f(x-2)\)，則（）A.\(f(x)\)的周期為\(4\)B.\(f(0)=f(4)\)C.\(f(1)=f(5)\)D.\(f(x)\)的圖象關于直線\(x=2\)對稱答案：ABC三、判斷題1.若函數(shù)\(f(x)\)滿足\(f(x+y)=f(x)+f(y)\)，則\(f(x)\)一定是正比例函數(shù)。（）答案：錯誤2.函數(shù)\(f(x)\)是定義在\(R\)上的奇函數(shù)，若\(f(0)\)有意義，則\(f(0)=0\)。（）答案：正確3.已知函數(shù)\(f(x)\)滿足\(f(x+2)=f(x)\)，則\(f(x)\)的圖象關于直線\(x=1\)對稱。（）答案：錯誤4.若函數(shù)\(f(x)\)是偶函數(shù)，且在\((0,+\infty)\)上單調遞增，則在\((-\infty,0)\)上單調遞減。（）答案：正確5.函數(shù)\(f(x)\)滿足\(f(x+1)=-f(x)\)，則\(f(x)\)的周期為\(1\)。（）答案：錯誤6.已知函數(shù)\(f(x)\)對于任意\(x_1\)，\(x_2\inR\)，當\(x_1\ltx_2\)時，都有\(zhòng)(f(x_1)\gtf(x_2)\)，則\(f(x)\)在\(R\)上單調遞減。（）答案：正確7.若函數(shù)\(f(x)\)滿足\(f(x)=f(-x)\)且\(f(x+2)=f(x)\)，則\(f(x)\)的圖象關于直線\(x=1\)對稱。（）答案：正確8.函數(shù)\(f(x)\)滿足\(f(x+4)=\frac{1}{f(x)}\)，則\(f(x)\)的周期為\(8\)。（）答案：正確9.已知函數(shù)\(f(x)\)在\([a,b]\)上單調遞增，在\([b,c]\)上單調遞減，則\(f(x)\)在\(x=b\)處取得最大值。（）答案：正確10.若函數(shù)\(f(x)\)滿足\(f(x+3)=f(x-1)\)，則\(f(x)\)的周期為\(4\)。（）答案：正確四、簡答題1.已知函數(shù)\(f(x)\)滿足\(f(x+y)=f(x)+f(y)\)，\(f(1)=2\)，求\(f(3)\)的值。答案：根據(jù)\(f(x+y)=f(x)+f(y)\)，令\(x=y=1\)，則\(f(2)=f(1+1)=f(1)+f(1)=2+2=4\)。再令\(x=2\)，\(y=1\)，那么\(f(3)=f(2+1)=f(2)+f(1)=4+2=6\)。2.函數(shù)\(f(x)\)是定義在\(R\)上的偶函數(shù)，且在\((0,+\infty)\)上單調遞減，比較\(f(-3)\)，\(f(2)\)，\(f(1)\)的大小。答案：因為\(f(x)\)是偶函數(shù)，所以\(f(-3)=f(3)\)。又因為\(f(x)\)在\((0,+\infty)\)上單調遞減，且\(3\gt2\gt1\gt0\)，所以\(f(3)\ltf(2)\ltf(1)\)，即\(f(-3)\ltf(2)\ltf(1)\)。3.已知函數(shù)\(f(x)\)滿足\(f(x+2)=-f(x)\)，且\(f(1)=3\)，求\(f(5)\)的值。答案：由\(f(x+2)=-f(x)\)可得\(f(x+4)=f((x+2)+2)=-f(x+2)=f(x)\)，所以函數(shù)\(f(x)\)的周期是\(4\)。那么\(f(5)=f(4+1)=f(1)\)，已知\(f(1)=3\)，所以\(f(5)=3\)。4.若函數(shù)\(f(x)\)滿足\(f(1-x)=f(1+x)\)，且\(f(x)\)在\((1,+\infty)\)上單調遞增，判斷\(f(x)\)在\((-\infty,1)\)上的單調性。答案：因為\(f(1-x)=f(1+x)\)，所以函數(shù)\(f(x)\)的圖象關于直線\(x=1\)對稱。已知\(f(x)\)在\((1,+\infty)\)上單調遞增，根據(jù)函數(shù)圖象的對稱性可知，\(f(x)\)在\((-\infty,1)\)上單調遞減。五、討論題1.已知函數(shù)\(f(x)\)是定義在\(R\)上的函數(shù)，滿足\(f(x+2)=f(x)\)且\(f(x)\)是偶函數(shù)，當\(x\in[0,1]\)時，\(f(x)=x\)，討論函數(shù)\(f(x)\)在\([-2,2]\)上的圖象與\(y=\frac{1}{2}\)的交點情況。答案：因為\(f(x+2)=f(x)\)，所以周期\(T=2\)，又\(f(x)\)是偶函數(shù)，\(x\in[0,1]\)時，\(f(x)=x\)。那么\(x\in[-1,0]\)時，\(f(x)=f(-x)=-x\)。在\([-2,-1]\)上，\(f(x)=f(x+2)=x+2\)；在\([1,2]\)上，\(f(x)=f(x-2)=-(x-2)=