NEW YORK — It was only her first day as an “American Idol” judge, but already Kara DioGuardi seemed part of the dysfunctional family.

DioGuardi, the freshly-installed fourth “Idol” critic, joined Simon Cowell, Paula Abdul and Randy Jackson in a public show of solidarity Tuesday afternoon as the quartet entered a Chelsea Piers building on Manhattan’s west side for a round of season eight auditions.

They walked a red carpet amid a media circus aimed at getting the first photo opportunity of DioGuardi with her new co-workers. Cowell summed up the thinking behind adding the 37-year-old singer-songwriter to the panel.

“It’s like, you know, having dinner with three people for seven years, and getting quite bored of each other,” Cowell told reporters. “So it’s having a fourth person, new opinion, I think it’s gonna be interesting.”

“I think it’s gonna make it sexier,” said Jackson, chiming in.

DioGuardi’s songs have been recorded by Kelly Clarkson, Christina Aguilera, Gwen Stefani, Celine Dion, Faith Hill, Carrie Underwood and Pink, among others. Her recent Top 40 hits include Stefani’s “Rich Girl,” Clarkson’s “Walk Away” and Dion’s “Taking Chances.”

Fox announced DioGuardi’s new gig Monday.

“Kara has a good reputation of A, having hits; and B, having an opinion. And that’s a good combination,” Cowell said, adding, matter-of-factly: “And she’s cute.”

“You think I’m cute?” DioGuardi replied. “Oh, good.”

A smiling Abdul said she was excited to have another woman at the judges’ table. She had expressed uncertainty about the new format in an radio interview the day before, telling Phoenix’s 104.7 KISS-FM that she was “concerned about the audience and acceptance. Time will tell. We’ll see.”

As it happens, Abdul and DioGuardi are longtime friends. They co-wrote the dance song “Spinning Around” for Kylie Minogue in 2000.

“I know Kara, and she speaks her mind — so she will be speaking her mind,” Abdul said of DioGuardi’s judging technique.

And how will the time-starved show fit in FOUR judges’ opinions?

“Paula will have to speak less,” said Cowell, unable to resist a swipe at his favorite target.