# Tagged Questions

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### submodular-like functions on $\mathbb{R}$

The definition of submodularity led me to define a type of function on numbers (I suppose ordered rings, but consider the reals for now) as $f(x) + f(y) \ge f(x+y) + f(xy)$. Such functions exist: for ...
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### Functional inequality and one identity

I'm a high school student from Bonn, Germany and I have to solve the following problem: If $g:R \rightarrow R$ is a function with the property $g(ab)-ag(b)\leq bg(a)$, for all real numbers a and b, ...
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### Finding range of transformation of function from range of original

I'm asked to find the range of $y = f(x-2)+4$, if the range of $y=f(x)$ is {$y| -2 \geq y \geq 5, y \in R$}. How do I go about finding this? I have no idea where to even start. I'm doing the course ...
### If $f(x)\leq f(f(x))$ for all $x$, is $x\leq f(x)$?
If I have $f(x)\leq f(f(x))$ for all real $x$, can I deduce $x\leq f(x)$? Thank you.
### Geometric intuition for the inequality $(f(y) - c) ( y - d ) \geq (f(d) - c) ( f^{-1}(c) - d )$
Good day to everyone. I am interested in the geometric intuition for the following statement: Let $f:\mathbb{R} \mapsto \mathbb{R}$ be a monotonically increasing, invertible function and \$c,d \in ...