The Portuguese signed a four-year contract with the Spanish giants following his move from European champions Inter Milan.
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"I like to give myself a challenge and this is a big one," Mourinho told a news conference at the Bernabeu Stadium. "I have a lot of confidence in myself and my ability as a coach.
"I want to thank the board for hiring me and I promise I will not change. I am Jose Mourinho and I arrive here with all my qualities and my defects," Mourinho added.
"The strength of my teams is the team not individual players or the coach. If they all recognise this it will not be difficult to get results."
Real director general Jorge Valdano opened proceedings by clearing the water over an article he wrote as a newspaper columnist criticising the way Mourinho's teams played a few years ago.
"Three years ago we sorted this out quietly, face to face. We don't have any problems with each other," the Argentine said sitting should-to-shoulder with Mourinho.
"We are delighted to have him here with us. The best thing for our club right now is to have Mourinho with us."
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The 47-year-old Mourinho, former manager of Porto and Chelsea, replaced Manuel Pellegrini who was sacked last week, becoming Real president Florentino Perez's eighth coach in seven years.
Pellegrini and coaches before him have complained about being isolated from decision-making at the club but Mourinho played down suggestions that he had demanded a greater say.
"I am a coach and nothing more," he said. "It's too early to speak about changes. I have to analyse the situation, ask many questions, listen to the answers, and then decide on a way forward."
Looking ahead to possible new signings, Mourinho signalled a change in direction in the type of players he would be seeking for the side.
"The club is happy with the base of the team. We need three or four players to give us flexibility in the way we might play, but I predict a quiet summer," he said.
"It's normal that people ask about players such as Maicon, Ashley Cole or Frank Lampard, for example, because they are former players of mine and I always have good relations with them.
"We need small adjustments. We have clear profiles of the ty
