If $X=f^{-1}(Y)$, is it true that $f(X)=Y$?
68746
What is $f(f^{-1}(A))$?
40070
Is there a real-valued function $f$ such that $f(f(x)) = -x$?
26677
thoughts about $f(f(x))=e^x$
88278
Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective
50655
Proof of $f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2})$
108923
What is wrong with this false proof? $-1=(-1)^1=(-1)^\frac{2}{2}=((-1)^2)^\frac{1}{2}=\sqrt{(-1)^2}=\sqrt{1}=1$
82097
If $A \succeq B$, is it true that $B^{-1} \succeq A^{-1}$?
6482
Why aren't the functions $f(x) = \frac{x-1}{x-1}$ and $f(x) = 1$ the same?
49705
Is $f(x,y)=(x+2y+y^2+|xy|, 2x+y+x^2+|xy|)$ differentiable?
21891
If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t
151519
If $f(f(x))=x^2-x+1$, what is $f(0)$?
81720
$f(x) + f(x+f(y)) = y + f(f(x) + f(f(y)))$
50725
Inverse of function is continuous or not?
32599
Solution to the functional equation $f(x^y)=f(x)^{f(y)}$
121395
Let $X$ and $Y$ be non-empty sets and let $f:X \rightarrow Y$. Prove that $ f^{ −1} :P(Y ) → P(X) $ is injective if, and only if, $f$ is surjective.
132113
Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
11453
Find a real function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?