The M1M1 treasure hunt, week 3

Ann O'Malley gazed at the plaque on the blue door, which read "Dr Hu." A little nervously, she knocked and entered a surprisingly spacious room.
"Ah, come in," a slightly dishevelled man greeted her. "I am The Doctor. You must be my new assistant or, as we now call them, personal tutee."
"Why," she exclaimed, "Your office is bigger on the inside than on the outside!"
"Yes, yes, everyone says that. Sorry about last week. I wasn't here."
"Where were you, Doctor?" she asked politely.
"Er, nowhere. I was somewhen else."
"I'm sorry?"
"Look, I travel a lot. Don't worry about it now, you understood it perfectly in a few weeks time. You said I should call you Ann."
"No I didn't, but you may."
"OK, Ann. Now why did you do so poorly in next week's test? Didn't you understand my explanations?"
"Er, no, I am a bit confused. You talk about the future as if it had already happened."
"Well it has for the people who were there, hasn't it? You will recall that I explained it was all about inverse functions."
"I'm sorry, I don't recall that," Ann replied meekly.
"You're not listening! I said you WILL recall that. How could you recall it before it happened?" snapped the Doctor tetchily.
"Now this office has a function, it takes us forward in time. But something, maybe the Universe, maybe Reality, maybe the College Health & Safety Dept. doesn't like that, and tries to reset us using the inverse function. But if they use the wrong one, instead of taking us back to when we started, it transports us somewhen else. You see?"
"Um, so what is the function of this office?" Ann asked carefully.
"A good question. If t is the fraction of this hourly tutorial which has passed and 1/2 < t < 1, then when I push this light switch, the room shifts to a new time, s, given by

s = f(t) = (t + (2t - 1)^1/2 )^1/2 − (t - (2t - 1)^1/2 )^1/2,

where the half-powers denote positive real numbers. I could have written "sqrt(2t-1)", or for integer powers "t^3". Anyway, the universe doesn't like that, and instantly resets the time using the inverse function

t = g(s) = ( ) /4 ?

So saying, The Doctor flipped the switch. The lights flickered for a moment. Nervously, Ann opened the door and peered out at the grim expressionless corridors of the Huxley Building - nothing had changed.
She looked at her watch, and saw that t was approaching 1; the strange tutorial was almost over. "Now," intoned the doctor. "What do you think happens if t > 1? What is s then?"

s = ?

"Yes! Exactly! Whatever t is! So now when the universe tries to send us back it doesn't know when we came from. So we get transported to a different time. With a little more effort, I can make it send us anywhere, anywhen."
"But that's irrational! You're saying that time-travel is possible because Reality makes mistakes?"
"Why not? Most things happen because of error rather than design. I mean look at your January Test answers! You need practice inverting functions. When you came to see me next week, you told me the wrong answer for the maximum value of a for which the function

f(x) = x³ + a x² + x + b was invertible."

a_max = ?