Just let me stay, even if its for a little while - the relationship
InuYasha first met Kagome when she arrived in the Fuedal Era and woke him from his magical sleep. At first, when InuYasha thought Kikyou had betrayed him, he was initially hurt and angry. He took it out on Kagome thinking she was Kikyou as they both look similar. Eventually, he realized that she was not Kikyou and slowly opened his heart. She was the first friend who showed compassion and kindness. Throughout the course of their journey, InuYasha and Kagome began to develop romantic feelings for each other and slowly began to fall in love, despite the numerous arguments and obstacles they faced such as saying her favorite word: Osuwari or Sit Boy!
Their friends, Sango and Miroku, were able to see how their relationship was starting to grow and started to talk with them. Shippou even tried talking about their relationship by causing friendly arguments with InuYasha to encourage him, even if it was to bring Kagome back to to the past. It appeared that everyone around them is aware of their strong romantic feelings for each other. This included Kagome's modern day friends and family to even their own enemies such as Naraku. Overall, throughout the series it is shown from the beginning to the end that they have come a long way from being close friends to falling strongly in love to the point where they become fiercely protective of each other.
As a result of his feelings towards her, InuYasha became fiercely protective of Kagome, saying that "the thought of losing two women is unbearable." InuYasha often competed with Kouga for Kagome's affections, though Kagome never showed any signs of attraction towards Kouga. She always kindly rejecting his advances towards her. InuYasha's love for Kagome was quite obvious as his friends immediately recognized his very strong and fierce protectiveness over Kagome. His enemies, including Naraku, recognized this and sometimes used Kagome to gain an advantage over InuYasha. Several times throughout the series when InuYasha was on the brink of death or gravely wounded, Inuyasha would spontaneously regain physical strength beyond all odds in order to protect Kagome, much to the surprise of his friends.
However, their relationship sometimes became strained and complicated due to Kikyou. Whenever Kikyou becomes involved somehow, InuYasha drops everything around him so that he may protect her. Events to which always lead to Kagome's feelings becoming deeply hurt. Kikyou has also been the source of many of Kagome and InuYasha's fights, Kagome has sometimes even considered leaving InuYasha as even though she is in love with him, it's too difficult for her to be with him knowing he still harbors feelings toward Kikyou. Kagome, however, has always been able to somehow learn to look past these doubts and always comes back to InuYasha's side.
Overall, despite the arguments and obstacles in their relationship, ranging from overbearing jealously to just pure stubbornness on both parts, InuYasha and Kagome's feelings for one another are strong and pure. The bond the two of them share shows that they would go to great lengths to protect each other or help in any way they can. As time goes on, InuYasha and Kagome are shown to learn to over come anything either it being simple arguments or rivals.