When is $f(f^{-1}(B)) \neq B$?
2072
Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$
97184
Show: $ f(a) = a,\ f(b) = b \implies \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $
122407
If $f(x) = x^3+3x+4\;,$ Then value of $ \int_{-1}^{1}f(x)dx+\int_{0}^{4}f^{-1}(x)dx$
6650
Why $\sqrt{-1 \times -1} \neq \sqrt{-1}^2$?
2414
If $f \in \operatorname{Hol}(D)$, $f(\frac{1}{2}) + f(-\frac{1}{2}) = 0$, prove that $|f(0)| \leq \frac{1}{4}$
51659
If $f(0)=0$ then prove that $|f(z)+f(-z)| \leq 2|z|^2$
76610
Show that $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$
35556
$f(z)=\frac{z}{1+|z|} $ Is 1-1?
88278
Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective
61799
Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
126698
measurability of a function -equivalent conditions
11453
Find a real function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?
140459
Why is $\left(e^{2\pi i}\right)^i \neq e^{-2 \pi}$?
13575
Need help for proving that: $f(f^{-1}(A)) ⊆ A$
13150
If the derivative of $f$ is never zero, then $f$ is one-to-one
56958
bounding a $C^1$ function that upholds $f'(t)\leq 1-f(t)^2$
110643
What are the strategies I can use to prove $f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$?