函數(shù)圖象高考真題及答案

一、單項(xiàng)選擇題1.函數(shù)\(y=\log_{2}(x+2)\)的圖象大致是（）A.B.C.D.答案：A2.函數(shù)\(f(x)=\frac{1-x}{1+x}\)的圖象關(guān)于（）A.\(y\)軸對(duì)稱B.直線\(y=x\)對(duì)稱C.坐標(biāo)原點(diǎn)對(duì)稱D.直線\(y=-x\)對(duì)稱答案：C3.函數(shù)\(y=x\cosx+\sinx\)的圖象大致為（）A.B.C.D.答案：D4.已知函數(shù)\(f(x)=2^{x}\)，\(g(x)=\log_{2}x\)，則下列結(jié)論中正確的是（）A.\(f(x)\)與\(g(x)\)的圖象關(guān)于\(y\)軸對(duì)稱B.\(f(x)\)與\(g(x)\)的圖象關(guān)于原點(diǎn)對(duì)稱C.\(f(x)\)與\(g(x)\)的圖象關(guān)于直線\(y=x\)對(duì)稱D.\(f(x)\)與\(g(x)\)的圖象關(guān)于直線\(y=-x\)對(duì)稱答案：C5.函數(shù)\(y=\frac{1}{x-1}\)在區(qū)間\([2,3]\)上的圖象的端點(diǎn)值為（）A.\(y=1\)，\(y=\frac{1}{2}\)B.\(y=0\)，\(y=1\)C.\(y=1\)，\(y=0\)D.\(y=\frac{1}{2}\)，\(y=0\)答案：A6.函數(shù)\(y=\sin(2x+\frac{\pi}{3})\)的圖象可由函數(shù)\(y=\sin2x\)的圖象（）A.向左平移\(\frac{\pi}{3}\)個(gè)單位長(zhǎng)度得到B.向右平移\(\frac{\pi}{3}\)個(gè)單位長(zhǎng)度得到C.向左平移\(\frac{\pi}{6}\)個(gè)單位長(zhǎng)度得到D.向右平移\(\frac{\pi}{6}\)個(gè)單位長(zhǎng)度得到答案：C7.若函數(shù)\(y=f(x)\)的圖象經(jīng)過(guò)點(diǎn)\((1,2)\)，則函數(shù)\(y=f(-x)+1\)的圖象必經(jīng)過(guò)點(diǎn)（）A.\((-1,3)\)B.\((1,3)\)C.\((-1,1)\)D.\((1,1)\)答案：A8.函數(shù)\(y=\frac{3x}{x^{2}+x+1}(x\lt0)\)的值域是（）A.\((-1,0)\)B.\([-3,0)\)C.\([-3,-1]\)D.\((-\infty,0)\)答案：B9.函數(shù)\(y=\log_{a}(x+3)-1(a\gt0,a\neq1)\)的圖象恒過(guò)定點(diǎn)\(A\)，若點(diǎn)\(A\)在直線\(mx+ny+1=0\)上，其中\(zhòng)(mn\gt0\)，則\(\frac{1}{m}+\frac{2}{n}\)的最小值為（）A.\(3+2\sqrt{2}\)B.\(3-2\sqrt{2}\)C.\(8\)D.\(4\)答案：C10.已知函數(shù)\(f(x)\)滿足\(f(x+1)=\frac{1+f(x)}{1-f(x)}\)，若\(f(1)=2\)，則\(f(2023)\)等于（）A.\(2\)B.\(-\frac{1}{3}\)C.\(-\frac{1}{2}\)D.\(3\)答案：C二、多項(xiàng)選擇題1.下列函數(shù)中，其圖象與函數(shù)\(y=\lnx\)的圖象關(guān)于直線\(x=1\)對(duì)稱的是（）A.\(y=\ln(1-x)\)B.\(y=\ln(2-x)\)C.\(y=\ln(1+x)\)D.\(y=\ln(2+x)\)答案：AB2.對(duì)于函數(shù)\(y=\sin(2x+\frac{\pi}{6})\)，下列說(shuō)法正確的是（）A.函數(shù)圖象關(guān)于點(diǎn)\((\frac{\pi}{3},0)\)對(duì)稱B.函數(shù)圖象關(guān)于直線\(x=-\frac{\pi}{6}\)對(duì)稱C.函數(shù)在區(qū)間\([-\frac{\pi}{6},\frac{\pi}{3}]\)上單調(diào)遞增D.函數(shù)圖象可由\(y=\sin2x\)的圖象向左平移\(\frac{\pi}{12}\)個(gè)單位長(zhǎng)度得到答案：ACD3.已知函數(shù)\(f(x)=a^{x}(a\gt0,a\neq1)\)，\(g(x)=\log_{a}x(a\gt0,a\neq1)\)，則下列說(shuō)法正確的是（）A.\(f(x)\)與\(g(x)\)的定義域和值域正好互換B.\(f(x)\)與\(g(x)\)的單調(diào)性相同C.\(f(x)\)與\(g(x)\)的圖象關(guān)于直線\(y=x\)對(duì)稱D.若\(f(x)\)過(guò)點(diǎn)\((2,4)\)，則\(g(x)\)過(guò)點(diǎn)\((4,2)\)答案：ABCD4.函數(shù)\(y=\frac{1}{x^{2}+1}\)的圖象具有的性質(zhì)是（）A.關(guān)于\(y\)軸對(duì)稱B.最大值為\(1\)C.在\((0,+\infty)\)上單調(diào)遞減D.是奇函數(shù)答案：ABC5.若函數(shù)\(y=f(x)\)的圖象上存在兩點(diǎn)，使得函數(shù)的圖象在這兩點(diǎn)處的切線互相垂直，則稱\(y=f(x)\)具有\(zhòng)(T\)性質(zhì)。下列函數(shù)中具有\(zhòng)(T\)性質(zhì)的是（）A.\(y=\sinx\)B.\(y=\lnx\)C.\(y=e^{x}\)D.\(y=x^{3}\)答案：A6.已知函數(shù)\(f(x)=|x-1|+|x+1|\)，則下列說(shuō)法正確的是（）A.\(f(x)\)的圖象關(guān)于\(y\)軸對(duì)稱B.\(f(x)\)的值域是\([2,+\infty)\)C.\(f(x)\)在\((-\infty,-1)\)上單調(diào)遞減D.\(f(x)\)在\((1,+\infty)\)上單調(diào)遞增答案：ABCD7.函數(shù)\(y=\log_{a}(x-1)+2(a\gt0,a\neq1)\)的圖象恒過(guò)定點(diǎn)\(P\)，則下列說(shuō)法正確的是（）A.點(diǎn)\(P\)的坐標(biāo)為\((2,2)\)B.當(dāng)\(a\gt1\)時(shí)，函數(shù)在\((1,+\infty)\)上單調(diào)遞增C.當(dāng)\(0\lta\lt1\)時(shí)，函數(shù)在\((1,+\infty)\)上單調(diào)遞減D.函數(shù)圖象與\(y=\log_{a}x\)的圖象形狀相同答案：ABCD8.已知函數(shù)\(y=f(x)\)的圖象是連續(xù)不斷的曲線，且有如下對(duì)應(yīng)值表：|\(x\)|\(1\)|\(2\)|\(3\)|\(4\)|\(5\)||---|---|---|---|---|---||\(f(x)\)|\(-2\)|\(3\)|\(-1\)|\(5\)|\(-3\)|則函數(shù)\(y=f(x)\)在區(qū)間（）內(nèi)至少有零點(diǎn)。A.\((1,2)\)B.\((2,3)\)C.\((3,4)\)D.\((4,5)\)答案：ABD9.下列函數(shù)中，在區(qū)間\((0,+\infty)\)上單調(diào)遞增的是（）A.\(y=x^{\frac{1}{2}}\)B.\(y=2^{x}\)C.\(y=\log_{2}x\)D.\(y=\frac{1}{x}\)答案：ABC10.已知函數(shù)\(f(x)\)是定義在\(R\)上的奇函數(shù)，當(dāng)\(x\gt0\)時(shí)，\(f(x)=x^{2}-2x\)，則下列說(shuō)法正確的是（）A.\(f(0)=0\)B.當(dāng)\(x\lt0\)時(shí)，\(f(x)=-x^{2}-2x\)C.函數(shù)\(f(x)\)的圖象關(guān)于原點(diǎn)對(duì)稱D.函數(shù)\(f(x)\)在\(R\)上的最大值為\(1\)答案：ABC三、判斷題1.函數(shù)\(y=\log_{a}x(a\gt0,a\neq1)\)的圖象一定過(guò)點(diǎn)\((1,0)\)。（）答案：對(duì)2.函數(shù)\(y=\sinx\)的圖象向右平移\(\frac{\pi}{2}\)個(gè)單位長(zhǎng)度得到\(y=\cosx\)的圖象。（）答案：錯(cuò)3.函數(shù)\(y=x^{2}\)與\(y=-x^{2}\)的圖象關(guān)于\(x\)軸對(duì)稱。（）答案：對(duì)4.若函數(shù)\(y=f(x)\)滿足\(f(a+x)=f(a-x)\)，則函數(shù)\(y=f(x)\)的圖象關(guān)于直線\(x=a\)對(duì)稱。（）答案：對(duì)5.函數(shù)\(y=\frac{1}{x}\)在其定義域上是單調(diào)遞減函數(shù)。（）答案：錯(cuò)6.函數(shù)\(y=a^{x}(a\gt0,a\neq1)\)與\(y=\log_{a}x(a\gt0,a\neq1)\)的圖象沒(méi)有交點(diǎn)。（）答案：錯(cuò)7.函數(shù)\(y=\sin(2x+\frac{\pi}{3})\)的最小正周期是\(\pi\)。（）答案：對(duì)8.函數(shù)\(y=\ln(x^{2}+1)\)的值域是\([0,+\infty)\)。（）答案：對(duì)9.函數(shù)\(y=3x+1\)的圖象是一條直線。（）答案：對(duì)10.若函數(shù)\(y=f(x)\)的圖象關(guān)于點(diǎn)\((a,b)\)對(duì)稱，則有\(zhòng)(f(a+x)+f(a-x)=2b\)。（）答案：對(duì)四、簡(jiǎn)答題1.簡(jiǎn)述函數(shù)\(y=\log_{a}x(a\gt0,a\neq1)\)的圖象性質(zhì)。答案：當(dāng)\(a\gt1\)時(shí)，函數(shù)\(y=\log_{a}x\)在\((0,+\infty)\)上單調(diào)遞增，圖象過(guò)定點(diǎn)\((1,0)\)，在\((0,1)\)上函數(shù)值小于\(0\)，在\((1,+\infty)\)上函數(shù)值大于\(0\)。當(dāng)\(0\lta\lt1\)時(shí)，函數(shù)在\((0,+\infty)\)上單調(diào)遞減，圖象同樣過(guò)定點(diǎn)\((1,0)\)，在\((0,1)\)上函數(shù)值大于\(0\)，在\((1,+\infty)\)上函數(shù)值小于\(0\)。2.如何由\(y=\sinx\)的圖象得到\(y=\sin(2x-\frac{\pi}{3})\)的圖象？答案：先將\(y=\sinx\)的圖象上所有點(diǎn)的橫坐標(biāo)縮短為原來(lái)的\(\frac{1}{2}\)（縱坐標(biāo)不變），得到\(y=\sin2x\)的圖象；再將\(y=\sin2x\)的圖象向右平移\(\frac{\pi}{6}\)個(gè)單位長(zhǎng)度，根據(jù)“左加右減”原則，就可得到\(y=\sin(2x-\frac{\pi}{3})\)的圖象。3.說(shuō)明函數(shù)\(y=x^{3}\)的圖象特點(diǎn)。答案：函數(shù)\(y=x^{3}\)是奇函數(shù)，圖象關(guān)于原點(diǎn)對(duì)稱。在\(R\)上單調(diào)遞增，當(dāng)\(x\lt0\)時(shí)，函數(shù)值小于\(0\)且函數(shù)圖象從左到右下降；當(dāng)\(x=0\)時(shí)，\(y=0\)；當(dāng)\(x\gt0\)時(shí)，函數(shù)值大于\(0\)且函數(shù)圖象從左到右上升，其圖象呈平滑的曲線狀。4.求函數(shù)\(y=2^{x-1}\)的反函數(shù)，并說(shuō)明其圖象與原函數(shù)圖象的關(guān)系。答案：由\(y=2^{x-1}\)可得\(x-1=\log_{2}y\)，即\(x=\log_{2}y+1\)，所以反函數(shù)為\(y=\log_{2}x+1(x\gt0)\)。原函數(shù)\(y=2^{x-1}\)與反函數(shù)\(y=\log_{2}x+1\)的圖象關(guān)于直線\(y=x\)對(duì)稱。五、討論題1.討論函數(shù)\(y=\frac{1}{x^{2}-1}\)的定義域、值域、單調(diào)性和奇偶性。答案：定義域：由\(x^{2}-1\neq0\)，得\(x\neq\pm1\)，即定義域?yàn)閈(\{x|x\neq\pm1\}\)。值域：令\(t=x^{2}-1\)，\(t\geq-1\)且\(t\neq0\)，則\(y=\frac{1}{t}\)，當(dāng)\(-1\leqt\lt0\)時(shí)，\(y\leq-1\)；當(dāng)\(t\gt0\)時(shí)，\(y\gt0\)，值域?yàn)閈((-\infty,-1]\cup(0,+\infty)\)。單調(diào)性：在\((-\infty,-1)\)和\((-1,0)\)上單調(diào)遞增，在\((0,1)\)和\((1,+\infty)\)上單調(diào)遞減。奇偶性：\(f(-x)=\frac{1}{(-x)^{2}-1}=\frac{1}{x^{2}-1}=f(x)\)，是偶函數(shù)。2.已知函數(shù)\(y=f(x)\)的圖象關(guān)于直線\(x=2\)對(duì)稱，且在\((2,+\infty)\)上單調(diào)遞增，討論\(f(1)\)與\(f(3)\)的大小關(guān)系，并說(shuō)明理由。答案：因?yàn)楹瘮?shù)\(y=f(x)\)的圖象關(guān)于直線\(x=2\)對(duì)稱，所以\(f(1)=f(3)\)。根據(jù)函數(shù)圖象的對(duì)稱性，點(diǎn)\((1,f(1))\)關(guān)于直線\(x=2\)的對(duì)稱點(diǎn)為\((3,f(1))\)，即\(f(1)\)和\(f(3)\)對(duì)應(yīng)的是函數(shù)圖象上關(guān)于直線\(x=2\)對(duì)稱的兩點(diǎn)的函數(shù)值