Given the function $$ f(x)=\frac{1}{4}\left((x-1)^{2}+7\right) $$
The first part of the question asks to find the largest domain containing the value $x=3$ for which $f^{-1}(x)$ exists. I determined the domain to be $x≥1$.
The second part of the question is:
Let $a$ be a real number not in the domain found in the previous part, find the exact value of $f^{-1}(f(a))$.
My thinking process was since $a<1$, based from the domain we found previously, then therefore $f(a)=f(-a)$.
Do I use the inverse function i.e. $f^{-1}(x)=1+\sqrt{4x+7}$ and just sub in $-a$? I'm not entirely sure if this is correct. Any help is greatly appreciated!
If the function were symmetric about $x=0$, then you would have $f(-x) = f(x)$. In your case, it's symmetric about $x=1$, so the relationship is instead $f(1-x) = f(1+x)$. Thus $$f^{-1}(f(1-x)) = f^{-1}(f(1+x)) = 1+x.$$
Now substitute $a=1-x$.