The right back, who conceded he was responsible for Joel Nouble’s fourth-minute goal that gave West Lothian the lead in the league encounter in Govan, also promised a reaction to the performance.
Yesterday, Giovanni van Bronckhorst’s players were booed off the field by their own fans for the second time in four days, following a 1-0 home win over Dundee in the Premier Sports Cup quarter-final on Wednesday night.
Rangers face Napoli in the Champions League in Italy on Wednesday night, and after four successive defeats in Group A, they will need to win or draw against the section leaders.
However, their captain remains determined to put a wretched run of form at home and abroad firmly in the past in the Stadio Diego Armando Maradona in midweek.
“We know that is not good enough,” he said. “I have to take the blame for the first goal. It is poor defending by myself. After that they sit in, but we have more than enough time in the game to rectify it.
“Our first-half performance wasn’t good enough. In the second half we changed things and tried to create chances. But rescuing a point is not good enough, especially at home. We are all very disappointed in the changing room.
“We have to look at ourselves. We had an honest chat in the changing room and we need to look at ourselves. We got a point which is never good enough given the standards we set at this club.
“We can only apologise to the fans who watched, but we have to put it right as a team. It is only the boys in the changing room who can change things. The boss sets us out and we have got to deliver. We have not done that and we have let down the fans today.
“We have to have a reaction in midweek though we know it will be a tough game. We need to come out fighting and show a real attitude.”
Asked about being booed by Rangers supporters at half-time and full-time, Tavernier said: “You never want to hear it, but our performance over the 90 minutes wasn’t good enough. They have got every right to boo us off coming off the pitch.”
The English defender, who insisted he is in perfect physical condition despite persistent speculation that he is carrying an injury, pledged to do his utmost to silence Rangers’ critics and get their bid to land the Scottish title back on track.