document.write( "Question 1208460: Determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain: f(x) = x + 1 + 1/x\r
\n" ); document.write( "\n" ); document.write( "I already found this function is neither even nor odd, but I need help with the other parts. \n" ); document.write( "

Algebra.Com's Answer #846881 by ikleyn(52778) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain: f(x) = x + 1 + 1/x
\n" ); document.write( "I already found this function is neither even nor odd, but I need help with the other parts.
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " It is great that you were able to move so far on your own.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "The domain is the set of all real numbers except of x= 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "You may plot the graphs of this function.\r\n" );
document.write( "\r\n" );
document.write( "For it, go to web-site  www.desmos.com/calculator\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Find there free of charge plotting tool for common use.\r\n" );
document.write( "\r\n" );
document.write( "Print there the formula for this function.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "You will see the plot immediately.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The plot consists of two branches. \r\n" );
document.write( "\r\n" );
document.write( "One branch is in the 1st quadrant, where x i greater than 0.\r\n" );
document.write( "\r\n" );
document.write( "You will see that this function f(x) is decreasing in the interval 0 < x < 1 \r\n" );
document.write( "and is increasing in the interval (1,oo).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The other branch is in the 3rd quarter, where x < 0.\r\n" );
document.write( "\r\n" );
document.write( "You see there that this function is increasing in the interval (-oo,-1) and is decreasing in the interval (-1,0).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, from the plots you have a huge amount of information.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next, you may get the same conclusions using formal considerations.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For it, take the derivative   f'(x) = 1 -  = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Use the standard recipes from Calculus:\r\n" );
document.write( "\r\n" );
document.write( "        The function is increasing where its derivative is positive.\r\n" );
document.write( "\r\n" );
document.write( "        The function is decreasing where its derivative is negative.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Having the derivative, you can analyze the behavior of the function in each \r\n" );
document.write( "of these 4 intervals, and you inevitably will come to the same conclusions LOGICALLY, to which you observed VISUALLY.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Happy making these standard exercises and have fun (!)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Enjoy the power of Calculus !\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "All these recipes of Calculus are just known for more than 400 years !\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The great minds of Newton and Leibniz were the great-fathers of the Calculus in the XVII century.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "They created Calculus for all of us and for you personally to make your spirit high and your vision far ahead.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );