Question 206433
A graph of {{{ (2x+1) / (x-7) }}} is {{{ graph(400,200,-70,25,-17,20,  (2x+1) / (x-7) ) }}}

As x gets larger and larger positively or negatively, y will exist (so x will also) and will get close to 2.  However, as x gets closer and closer to 7 the numerator gets closer and closer to 15, but the denominator gets closer and closer to 0, meaning the denominator is tinier and tinier.  But, if you take a number near 15 and divide by a tiny number you get a large (+ or -) number.  For instance, if x = 6.99 then the numerator is 14.98 and the denominator is -0.01 which is the same as -1/100.  
So, {{{ g(6.99) = 14.98/-0.01 = 14.98*-100 =-1498 }}} which is a huge number, way off the bottom of the graph and {{{ g(7.01) = 15.02/0.01 = 15.02*100 =1502}}} which is a huge number, way off the top of the graph.

Speaking of the graph, near to x = 7, you can see part of the graph that is a vertical line. That shouldn't actually be there. Near to 7 the actual graph just continues farther and farther off the bottom and top of the picture with nothing in the middle (in the picture, where x very near to 7).  If x = 7 then g(7) is 15 / 0, but, since you can't divide by 0, g(7) does not exist, so 7 is not in the domain of g.  All this is to say that the domain of g is ALL REAL NUMBERS EXCEPT FOR 7.
   -  Mick