1 . 已知直线
与反比例函数
的图象交于
,
两点.
(1)求反比例函数的解析式及点B的坐标.
(2)当
时,则x的取值范围是______.
(3)连接
并延长与第一象限的双曲线交于点C,连接
、
,请直接写出
的面积与
的面积之间的数量关系.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41d49bb2fdf76bbf030f0749462ab208.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6db8dc0564b187f7adb8184f714ab5da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0778813665f307942db9769077032f97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb2e8d405145573838999a8695cb8332.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/17/04c467b0-e9c2-4e90-aaaf-31896b1b22ca.png?resizew=168)
(1)求反比例函数的解析式及点B的坐标.
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f99df1a7b58018125b99578b779342.png)
(3)连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf1438142deeac876fc7dc50552e552.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/173bc7a36757d77f01213411edd25241.png)
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2 . 阅读与思考
阅读下列材料,并按要求完成相应的任务.
探究反比例函数图象中的等线段
我们知道,若反比例函数
的图象与正比例函数
的图象相交于点
,
,则根据反比例函数的图象与正比例函数的图象都关于原点对称,不难发现
,那么如果反比例函数
的图象与一次函数
的图象相交于点
,
,一次函数的图象与
轴,
轴分别交于点
,
,是否也存在相等线段?
下面分别从反比例函数图象与一次函数图象的交点在同一象限和不同象限两种情况进行分析:
情况
:交点在同一象限(以交点在第一象限为例).
如图
,过点
作
轴于点
,作
轴于点
,
,
交于点
,连接
.
设点
,
,
则
,
,
,
,
,
,
.
又
,
∽
(依据),
,
.
又
,
四边形
是平行四边形,
.
同理可得
,从而
;
情况
:交点在不同象限(以交点在一、三象限为例).
如图
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07096af3b99fd1cb11c31f19a2c6408e.png)
任务:
(1)上述证明过程中的依据是:______ ;
(2)请参照情况
的分析过程,写出情况
的分析过程;
(3)“从一般到特殊”的思想拓展研究数学中的一些问题,是数学中经常使用的解题方法,结合以上信息,猜想:当反比例函数
的图象与一次函数
的图象只有
个交点时,设交点为
,一次函数
的图象与
轴,
轴分别交于点
,
,试着找出一条结论:______ .
阅读下列材料,并按要求完成相应的任务.
探究反比例函数图象中的等线段
我们知道,若反比例函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07854693dd2e33f66030d6106eb6e0ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cca39b30b0b8e769293e13546b91f35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fca3734de79f7f50b552ef62b29dc7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07854693dd2e33f66030d6106eb6e0ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89c1972246a0a3d1c987d25205dbdd99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
下面分别从反比例函数图象与一次函数图象的交点在同一象限和不同象限两种情况进行分析:
情况
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e1c9ae241fd78126274c65e17990c88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d9ef2b0fd5b74974b9db2009c597af0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/715d55d3441ce4df008c4d7ca4547ac1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2736043b87175f0cf8e2502539da8cf5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/063e6a535f69b40f703caad0d349eb9f.png)
则
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb8382d5c1ccc3caadfc092b5829cfbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d450542699c52717f078d3077161990e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fd686cd4d7a8e1f7b6c85c244ef18d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51a20332366d18c3415248fb4cc8f240.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddec4f02b95e9f4fe5edd404ec232785.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4286feda54ba56e4258e3d20f285ffd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a1724ba38ce0fad5f1acfddcbbb4e39.png)
又
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39d96c97ab2ab7f0c353f3dde459a23b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bbd8b433480a36188a6710e70c28151.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d513f3f07ebd929e2fdd077025c76140.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/addeb3b3993e2252b5c1355f1541134f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0162e7ebcb8c7d916de6b865799d91f4.png)
又
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e355c60f6dd643e8b3dd6aaa690aea28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4465a96d65de0f859ea4ec9e548204d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30fa0ee94853863494b1a71f3eb89526.png)
同理可得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9332b9191d830338ed51f96422879f03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7de966c316db1013defc56372fcf814e.png)
情况
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c650fe55b7603f106c53ca2423451c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07096af3b99fd1cb11c31f19a2c6408e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/25/752b489b-a347-4f4d-a7b6-8d8c9b8268c4.png?resizew=393)
任务:
(1)上述证明过程中的依据是:______ ;
(2)请参照情况
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
(3)“从一般到特殊”的思想拓展研究数学中的一些问题,是数学中经常使用的解题方法,结合以上信息,猜想:当反比例函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07854693dd2e33f66030d6106eb6e0ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89c1972246a0a3d1c987d25205dbdd99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89c1972246a0a3d1c987d25205dbdd99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
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3 . 如图,一次函数
的图像与反比例函数
的图像在第一象限交于点
,在第三象限交于点
,与
轴交于点
,过点
作
轴于点
,且
的面积是
.
(1)求一次函数和反比例函数的表达式.
(2)求
的面积.
(3)请直接写出不等式
的解集.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e114459637a3db306e91b5e9ee9aae8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f14938f867bb928aca6a28c6407995b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7742e18d0d1458c4837195ecda64db06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e3b6ff59b12a0d55fa2dc8cc0f34fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2241bd71e0c20867029e489f9ba655d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcb49df05f2e31d005735c3f14a21d30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8da45c443af7994a26ffa9d8894e7262.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/30/60b9db58-a8c4-496e-8c0b-b0b0d80ddd11.png?resizew=237)
(1)求一次函数和反比例函数的表达式.
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbfa1a2af7e38d33634c462300df381f.png)
(3)请直接写出不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83dcc99ba5edf3441f79356a527c6a4b.png)
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4 . 已知反比例函数
的图象与一次函数
的图象交于
两点,它们的横坐标分别为
,则关于
的不等式
的解为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ff4dc731bb311e88109839f15e69478.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fceb9eb57ca36db386ffafeda213599c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af88707991c079b07739feb16b0bf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efa9c3906e02f6e348ffb8b566403329.png)
A.![]() | B.![]() ![]() |
C.![]() | D.![]() ![]() |
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5 . 如图,在平面直角坐标系中,一次函数的图象与反比例函数的图象交于第二、四象限内的A,B两点,与x轴交于点C,与y轴交于点D,点B的坐标是
,点A的坐标是
.
(1)求反比例函数和一次函数的解析式;
(2)连接
,求
的面积.
(3)直接写出一次函数值大于反比例函数值时x的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/833bf886b9b2fc3745c170904aefe6d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae46d13b6282701f48e5c5f329a6c65a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/22/83970f4c-89dc-4a6a-aeaa-a09eea263acd.jpg?resizew=162)
(1)求反比例函数和一次函数的解析式;
(2)连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
(3)直接写出一次函数值大于反比例函数值时x的取值范围.
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6 . 在平面直角坐标系
中,一次函数
与y轴交于点C,与反比例函数
的图象交于A,B两点,点A的坐标为
.
(1)求一次函数与反比例函数的表达式;
(2)若y轴上存在点P,使得
是以
为腰的等腰三角形,请直接写出符合条件的点P的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbe3f062467b22028065486244bec1df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da60a36461ce314d34ad72593c59d1a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8ce2830528c2ea6b5d4df0c77644e33.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/14/79435de1-ab95-49ba-b7fa-98f2fbcf037b.png?resizew=179)
(1)求一次函数与反比例函数的表达式;
(2)若y轴上存在点P,使得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/147de24f071e316b68fd2e78e3c84545.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
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名校
7 . 如图,一次函数
的图象与反比例函数
的图象交于点
,
两点,当
时,
的取值范围是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/715416015a9634f5eafe3d399987d837.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d105b74c40724a7da626e7c42afbf5fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c61ba0aa60e487127dc2f53df86a854f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c264b929c9e1c61b86503977d7f33bb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f99df1a7b58018125b99578b779342.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/24/a49f59e3-e612-4679-ba14-2e28172f66b9.png?resizew=230)
A.![]() ![]() | B.![]() ![]() |
C.![]() ![]() | D.![]() ![]() |
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8 . 如图,一次函数
的图象与反比例函数
的图象交于
、
两点,当
时,
的取值范围是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd3c5db8861e2f0bb7848d0e4fe2da0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97fe8360ae73fc958662e5c517e28223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e6c8170dd3ad55d366c866dfa21c7df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a30ca15af0d5cbd9c45b2453a733810b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df86b0da538701c08fb214608e062372.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/20/f3d0977d-ba8f-4801-97f5-46ceceafcbe5.png?resizew=181)
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9 . 综合与实践
如图,一次函数
的图象与x轴交于点A,与y轴交于点B,把线段
绕点B逆时针旋转
得到
,过点C作
轴于点D,反比例函数
的图象经过点C,与直线
交于两点E和F.
(1)求反比例函数的解析式;
(2)如图2,若点E的横坐标是1,点F的纵坐标是
.
①直接写出线段
和
的数量关系和当
时,x的取值范围;
②连接
和
,求
的面积;
(3)当点M在x轴上运动,点N在反比例函数
的图象上运动,以点A,D,M和N为顶点的四边形是平行四边形,直接写出点M的坐标.
如图,一次函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6a4b66f74a71ee292a2b01f7ffccac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c02b54dc6b3e1bb6544f47d4c8743fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b4facfc5447ae7f665e8ec050b81313.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d105b74c40724a7da626e7c42afbf5fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/24/c41c5548-7118-4163-8912-7b68f7960290.png?resizew=488)
(1)求反比例函数的解析式;
(2)如图2,若点E的横坐标是1,点F的纵坐标是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81fb134b2b48acc99213fff6ccfee65f.png)
①直接写出线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e1113e9b568fb9ca3286582368e257a.png)
②连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35e5a933a2a0ff3a28009cc989293ff5.png)
(3)当点M在x轴上运动,点N在反比例函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d105b74c40724a7da626e7c42afbf5fa.png)
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3卷引用:山西省大同市云冈区2022-2023学年九年级上学期12月月考数学试题
山西省大同市云冈区2022-2023学年九年级上学期12月月考数学试题 山西省忻州市原平农业学校等2校2022-2023学年九年级上学期12月月考数学试题(已下线)专题6.30 反比例函数(全章分层练习)(培优练)-2023-2024学年九年级数学上册基础知识专项突破讲与练(北师大版)
10 . 如图,在平面直角坐标系中,反比例函数
与一次函数
的图象相交于点
与点
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/18/5b8f380d-f1c4-4f51-ad91-a6aa15928d99.png?resizew=196)
(1)求一次函数和反比例函数的表达式;
(2)求
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07854693dd2e33f66030d6106eb6e0ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ea3596cd249f618d60472ec5aeb609c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/921502954d8f4c6e58a95487018a8a04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6419152065edb8cabf887b65adb4a73.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/18/5b8f380d-f1c4-4f51-ad91-a6aa15928d99.png?resizew=196)
(1)求一次函数和反比例函数的表达式;
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
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4卷引用:山西省大同市2022-2023学年九年级上学期2月期末数学试题
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