She's correct in saying that the inverse relation of a function may not be a function itself. Every function has a corresponding inverse relation; however, we usually say a function is invertible iff its inverse relation is a function. However, the question specifically asked to find f^-1(x), which implies that the inverse relation is a function. After all, the letter f stands for function and any good book will define the notation f^-1 to stand for the inverse function of f. So for the question to ask for an inverse function of f(x) = x^2 - 6 requires a domain restriction. If the question asked to find the inverse, then I could see her point but since the question specifically used the notation f^-1(x), I completely agree with you.
In higher mathematics, there are such objects called multivalued functions, but we certainly don't study these concepts in any detail in high school.
The following Wikipedia article does the topic justice, and introduces multi-valued inverses:
It's the same concept in question 19. y = cos^-1(x) denotes the inverse cosine FUNCTION, which is the inverse of y = cos x restricted to a domain of [0, pi].
What's sad is that we're the concerned, diligent math teachers who want to get it right and you bring a legitimate concern to light and are told that you won't "win the argument". At the minimum the question is ambiguous and poorly posed, regardless of whether SED thinks you didn't win the argument. What's even sadder is that this test is supposedly reviewed by trained eyes. How are mathematically educated people not picking up on these glaring errors and ambiguities? I saw the ambiguities within seconds of reading the questions (as I'm sure most all of us did) and somehow those questions ended up on the test.