Relations and Functions

Relations and Functions

Question

Verify that  

space space space space space space space space space space space space space top enclose open parentheses fraction numerator 3 plus 4 straight i over denominator 1 minus 2 straight i end fraction close parentheses end enclose space equals space fraction numerator top enclose open parentheses 3 plus 4 straight i close parentheses end enclose over denominator top enclose open parentheses 1 minus 2 straight i close parentheses end enclose end fraction

Answer
space space space space space space space fraction numerator 3 plus 4 straight i over denominator 1 minus 2 straight i end fraction equals left parenthesis 3 plus 4 straight i right parenthesis fraction numerator left parenthesis 1 plus 2 straight i right parenthesis over denominator left parenthesis 1 right parenthesis squared plus left parenthesis 2 right parenthesis squared end fraction                   space space space space space space space space space space space open parentheses because fraction numerator 1 over denominator straight a plus ib end fraction equals fraction numerator straight a minus ib over denominator straight a squared plus straight b squared end fraction close parentheses

space space space space space space space space space space space equals space space space fraction numerator left parenthesis 3 plus 4 straight i right parenthesis left parenthesis 1 plus 2 straight i right parenthesis over denominator 5 end fraction equals fraction numerator 3 plus 10 straight i plus 8 straight i squared over denominator 5 end fraction equals fraction numerator negative 5 plus 10 straight i over denominator 5 end fraction equals negative 1 plus 2 straight i
rightwards double arrow      space space space space space space space top enclose open parentheses fraction numerator 3 plus 4 straight i over denominator 1 minus 2 straight i end fraction close parentheses end enclose space equals space top enclose open parentheses negative 1 plus 2 straight i close parentheses end enclose space equals space minus 1 minus 2 straight i                                                   ....(i)
Also,      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>
     space space space space space space equals space fraction numerator negative 5 minus 10 straight i over denominator 5 end fraction equals negative 1 minus 2 straight i                                                                       ....(ii)
Hence, from (i) and (ii),    space space space space space space space space space space space space space space space space space space space space top enclose open parentheses fraction numerator 3 plus 4 straight i over denominator 1 minus 2 straight i end fraction close parentheses end enclose space equals space top enclose fraction numerator open parentheses 3 plus 4 straight i close parentheses over denominator top enclose open parentheses 1 minus 2 straight i close parentheses end enclose end fraction end enclose

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