I liceum

I technikum

Matematyka 1. Zakres rozszerzony. Reforma 2019

Matematyka

MATeMAtyka 1. Karty pracy ucznia zakres podstawowy. Reforma 2019

Matematyka 1. Maturalne karty pracy zakres podstawowy cz. 1. Reforma 2019

Matematyka 1. Maturalne karty pracy zakres podstawowy cz. 2. Reforma 2019

MATeMAtyka 1. Maturalne karty pracy zakres podstawowy i rozszerzony. Reforma 2019

Matematyka 1. Maturalne karty pracy zakres rozszerzony cz. 1. Reforma 2019

Matematyka 1. Poziom podstawowy i rozszerzony. Po gimnazjum

Matematyka 1. Poziom podstawowy. Po gimnazjum

Matematyka 1. Zakres podstawowy i rozszerzony. Po gimnazjum

MATeMAtyka 1. Zakres podstawowy i rozszerzony. Po gimnazjum

MATeMAtyka 1. Zakres podstawowy i rozszerzony. Reforma 2019

Matematyka 1. Zakres podstawowy. Po gimnazjum

MATeMAtyka 1. Zakres podstawowy. Po gimnazjum

Matematyka 1. Zakres podstawowy. Reforma 2019

MATeMAtyka 1. Zakres podstawowy. Reforma 2019

Matematyka 1. Zakres rozszerzony. Cześć 1. Reforma 2019

Matematyka 1. Zakres rozszerzony. Cześć 2. Reforma 2019

Matematyka 1. Zakres rozszerzony. Po gimnazjum

Matematyka i przykłady jej zastosowań 1. Zakres podstawowy i rozszerzony. Reforma 2019

Matematyka i przykłady jej zastosowań 1. Zakres podstawowy. Reforma 2019

Matematyka poznać, zrozumieć 1. Zakres podstawowy i rozszerzony. Po gimnazjum

Matematyka poznać, zrozumieć 1. Zakres podstawowy. Po gimnazjum

Matematyka poznać, zrozumieć 1.Zakres podstawowy. Po gimnazjum

Matematyka. Zbór zadań maturalnych. Poziom podstawowy. Reforma 2019

Matematyka z plusem 1. Ćwiczenia podstawowe. Reforma 2019

Matematyka z plusem 1. Zakres podstawowy i rozszerzony. Reforma 2019

Matematyka z plusem 1. Zakres podstawowy. Reforma 2019

Matematyka z plusem 1. Zakres rozszerzony. Reforma 2019

Matura z Matematyki  2018 - ... Andrzej Kiełbasa. Poziom podstawowy cz. 1

Matura z Matematyki 2018 - ... Andrzej Kiełbasa. Poziom podstawowy cz. 2

Matura z Matematyki 2018 - ... Andrzej Kiełbasa. Poziom podstawowy i rozszerzony cz. 1

Matura z Matematyki 2018 - ... Andrzej Kiełbasa. Poziom podstawowy i rozszerzony cz. 2

Prosto do matury 1. Zakres podstawowy i rozszerzony. Po gimnazjum

Prosto do matury 1. Zakres podstawowy i rozszerzony. Reforma 2019

Prosto do matury 1. Zakres podstawowy. Po gimnazjum

Prosto do matury 1. Zakres podstawowy. Reforma 2019

Teraz matura. Matematyka. Zbiór zadań poziom podstawowy. Reforma 2019

Teraz matura. Matematyka. Zbiór zadań poziom rozszerzony. Reforma 2019

a)
1) Jest to punkt (0,0).
2) Jest to punkt (0,0) oraz punkt (2,0).

Zauważmy, że: 3−8<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><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'/></svg>​=−2 &nbsp; log2​16=4 &nbsp; &nbsp;&nbsp;

Obliczamy wartości funkcji&nbsp;f&nbsp;dla wszystkich argumentów ze zbioru&nbsp;X: f(3)=3−3<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><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'/></svg>​=0<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><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'/></svg>​=0 &nbsp;

a) Wykres funkcji f: 
<img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/85099/KphvLNxV0wVblKk1enKFKw%3D%3D_14e9a9ed49bf0a3d2f2b91f53bab469e4c974903a0458b7f1161e5a447b3bc30.jpg" alt="" width="538" height="235">

Wykres funkcji f: <img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/85145/FN%2BSBrKbgKlP09Ffgvr45w%3D%3D_b2bec742f37a5eabc467871e79ddcce9f8e30ef726f4e586c3b8da29d33cd19c.jpg">&nbsp; &nbsp;

a) <table style="border-collapse: collapse;width: 570px;height: 75px;" border="1"> <tbody> <tr> <td style="width: 78px;"> x &nbsp;</td> <td style="width: 78px;"> −3 &nbsp;</td> <td style="width: 78px;"> −2 &nbsp;</td> <td style="width: 78px;"> −1 &nbsp;</td> <td style="width: 78px;"> 0 &nbsp;</td> <td style="width: 78px;"> 1 &nbsp;</td> <td style="width: 78px;"> 4 &nbsp;</td> <td style="width: 78px;"> 9 &nbsp; </td> </tr> <tr> <td style="width: 78px;"> f(x) &nbsp;</td> <td style="width: 78px;"> 1 &nbsp;</td> <td style="width: 78px;"> 1 &nbsp;</td> <td style="width: 78px;"> 1 &nbsp;</td> <td style="width: 78px;"> 1 &nbsp;</td> <td style="width: 78px;"> −1 &nbsp;</td> <td style="width: 78px;"> −2 &nbsp;</td> <td style="width: 78px;"> −3 &nbsp;</td> </tr> </tbody> </table>

Zauważmy, że: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/8b674bb78eba8dbe88d665b8c03652754565fb3626746c5b6ca77df4d67e55b8_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/8b674bb78eba8dbe88d665b8c03652754565fb3626746c5b6ca77df4d67e55b8_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/c7ebe4c9826b2729c7c851c18d7cb939033fbe5b4b40f35cc0a5fbc29a2d4add_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/c7ebe4c9826b2729c7c851c18d7cb939033fbe5b4b40f35cc0a5fbc29a2d4add_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp;&nbsp;

Obliczamy wartości funkcji&nbsp;f&nbsp;dla wszystkich argumentów ze zbioru&nbsp;X: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/f490d86b3be6974ad5acd8425d30f067642dd0d6813d23169ffaf2185d725700_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/f490d86b3be6974ad5acd8425d30f067642dd0d6813d23169ffaf2185d725700_big.svg" type="image/svg+xml"></object> &nbsp;

Zauważmy, że: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/55af75354be1ff8ba58f3d7f149e0cf45da6a14dbcd065d3df74b2020fbbb02a_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/55af75354be1ff8ba58f3d7f149e0cf45da6a14dbcd065d3df74b2020fbbb02a_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/928e7175f1e3b0e301cd6e500c49ae893da771e628e4786c307aea2abee8e26c_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/928e7175f1e3b0e301cd6e500c49ae893da771e628e4786c307aea2abee8e26c_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/4c16985ff1808bda96ffc0712f85470c82121dce69696ec7c5daea2a5086b30f_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/4c16985ff1808bda96ffc0712f85470c82121dce69696ec7c5daea2a5086b30f_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/0be605bd139145386b15973e3ad98c9456b31a161a52a8934c9f6ef8b69d80e9_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/0be605bd139145386b15973e3ad98c9456b31a161a52a8934c9f6ef8b69d80e9_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/366469f7e802577a1bd490b0d7a4525a51e043e612503df8d3a1198f0e4120cb_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/366469f7e802577a1bd490b0d7a4525a51e043e612503df8d3a1198f0e4120cb_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; Zatem otrzymujemy: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/d4167b0322733b05db0c295910643e5fe0cc847959b2b8f3764ea8dfd910f917_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/d4167b0322733b05db0c295910643e5fe0cc847959b2b8f3764ea8dfd910f917_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/0e78b7fc94b4a7565ca9fafa9de7f014de403139740dacb05de422e590f48f8b_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/0e78b7fc94b4a7565ca9fafa9de7f014de403139740dacb05de422e590f48f8b_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/8f80719b30d718ef5aefd93edd02bd158e2325559ce8156c73453a46e753e1d5_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/8f80719b30d718ef5aefd93edd02bd158e2325559ce8156c73453a46e753e1d5_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/dbf8dfdaca4aeaa1acc5888af2bad6173f0f983871e4f4e83c3ebc3a62f8fc3b_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/dbf8dfdaca4aeaa1acc5888af2bad6173f0f983871e4f4e83c3ebc3a62f8fc3b_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/c28f9db1a68dbb5be8a0ee76ca04a30cac6bd9fbb5eb424d31dfc826575049d8_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/c28f9db1a68dbb5be8a0ee76ca04a30cac6bd9fbb5eb424d31dfc826575049d8_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/e03fa96478ad0e8ca4c12fb8784920c1c51da4e98a2dadce8f74ade6a038a253_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/e03fa96478ad0e8ca4c12fb8784920c1c51da4e98a2dadce8f74ade6a038a253_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; Wykres funkcji f: <img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/85098/YYxobdM%2B7lqP51RQwvnUIg%3D%3D_db0ceb42bdeb4c927ceb4d11f4da7b52db71b0dda8d8a275930a1e8fc0f8f44b.jpg" alt="" width="298" height="463">

a) <table style="border-collapse: collapse;width: 570px;height: 75px;" border="1"> <tbody> <tr> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/2d711642b726b04401627ca9fbac32f5c8530fb1903cc4db02258717921a4881_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/2d711642b726b04401627ca9fbac32f5c8530fb1903cc4db02258717921a4881_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/615bdd17c2556f82f384392ea8557f8cc88b03501c759e23093ab0b2a9b5cd48_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/615bdd17c2556f82f384392ea8557f8cc88b03501c759e23093ab0b2a9b5cd48_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/cf3bae39dd692048a8bf961182e6a34dfd323eeb0748e162eaf055107f1cb873_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/cf3bae39dd692048a8bf961182e6a34dfd323eeb0748e162eaf055107f1cb873_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/1bad6b8cf97131fceab8543e81f7757195fbb1d36b376ee994ad1cf17699c464_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/1bad6b8cf97131fceab8543e81f7757195fbb1d36b376ee994ad1cf17699c464_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/5feceb66ffc86f38d952786c6d696c79c2dbc239dd4e91b46729d73a27fb57e9_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/5feceb66ffc86f38d952786c6d696c79c2dbc239dd4e91b46729d73a27fb57e9_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/6b86b273ff34fce19d6b804eff5a3f5747ada4eaa22f1d49c01e52ddb7875b4b_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/6b86b273ff34fce19d6b804eff5a3f5747ada4eaa22f1d49c01e52ddb7875b4b_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/4b227777d4dd1fc61c6f884f48641d02b4d121d3fd328cb08b5531fcacdabf8a_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/4b227777d4dd1fc61c6f884f48641d02b4d121d3fd328cb08b5531fcacdabf8a_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/19581e27de7ced00ff1ce50b2047e7a567c76b1cbaebabe5ef03f7c3017bb5b7_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/19581e27de7ced00ff1ce50b2047e7a567c76b1cbaebabe5ef03f7c3017bb5b7_big.svg" type="image/svg+xml"></object> &nbsp; </td> </tr> <tr> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/259072e6c653cc56ea0e6a6031c2bbb5dede2c4066d63855fd7690ee8bd61bbc_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/259072e6c653cc56ea0e6a6031c2bbb5dede2c4066d63855fd7690ee8bd61bbc_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/6b86b273ff34fce19d6b804eff5a3f5747ada4eaa22f1d49c01e52ddb7875b4b_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/6b86b273ff34fce19d6b804eff5a3f5747ada4eaa22f1d49c01e52ddb7875b4b_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/6b86b273ff34fce19d6b804eff5a3f5747ada4eaa22f1d49c01e52ddb7875b4b_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/6b86b273ff34fce19d6b804eff5a3f5747ada4eaa22f1d49c01e52ddb7875b4b_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/6b86b273ff34fce19d6b804eff5a3f5747ada4eaa22f1d49c01e52ddb7875b4b_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/6b86b273ff34fce19d6b804eff5a3f5747ada4eaa22f1d49c01e52ddb7875b4b_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/6b86b273ff34fce19d6b804eff5a3f5747ada4eaa22f1d49c01e52ddb7875b4b_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/6b86b273ff34fce19d6b804eff5a3f5747ada4eaa22f1d49c01e52ddb7875b4b_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/1bad6b8cf97131fceab8543e81f7757195fbb1d36b376ee994ad1cf17699c464_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/1bad6b8cf97131fceab8543e81f7757195fbb1d36b376ee994ad1cf17699c464_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/cf3bae39dd692048a8bf961182e6a34dfd323eeb0748e162eaf055107f1cb873_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/cf3bae39dd692048a8bf961182e6a34dfd323eeb0748e162eaf055107f1cb873_big.svg" type="image/svg+xml"></object> &nbsp;</td> <td style="width: 78px;"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/615bdd17c2556f82f384392ea8557f8cc88b03501c759e23093ab0b2a9b5cd48_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/615bdd17c2556f82f384392ea8557f8cc88b03501c759e23093ab0b2a9b5cd48_big.svg" type="image/svg+xml"></object> &nbsp;</td> </tr> </tbody> </table>

$x$	$- 3$	$- 2$	$- 1$	$0$	$1$	$4$	$9$
$f (x)$	$1$	$1$	$1$	$1$	$- 1$	$- 2$	$- 3$

Komentarze