15 questions linked to/from half iterate of $x^2+c$
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### How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?

How to get $f(x)$, if we know that $f(f(x))=x^2+x$? Is there an elementary function $f(x)$ that satisfies the equation?
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### Find $f(x)$ such that $f(f(x)) = x^2 - 2$

Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$. Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but the ...
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### Find $f(f(\cdots f(x)))=p(x)$

$\newcommand{\nest}{\operatorname{nest}}$Let's define a function $\nest(f, x, k)$, which takes a function $f$, an input $x$, and a non-negative integer $k$, and calls $f$ on $x$ repeatedly ($k$ times)....
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### Find $f(x)$ satisfying $f(f(x))=x^x$

By inspection my attempts are always wrong. I really have no idea and given up. How to find $f(x)$ satisfying $f(f(x))=x^x$? My attempts: $f(x)=x^x$ $f(x)=x^{1/x}$ $f(x)=\frac{1}{x^x}$ My ...
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### How to solve the iterated function $f(f(x))=x^2+x$? [duplicate]

Any given setting for $f$ is acceptable. Iterated function
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### Find $f(0)$ if $f(f(x))=x^2-x+1$

Given $f : \mathbb{R} \to \mathbb{R}$ defined as $$f(f(x))=x^2-x+1$$ Find value of $f(0)$ I assumed $g(x)=f(f(x))$ we gave $$g(x)=(x-1)^2+(x-1)+1$$ Also $$g(x-1)=(x-1)^2-(x-1)+1$$ subtracting ...
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### find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that : $f(f(x))=x^2-2$ [duplicate]

This is a very hard functional equation. the problem is this : find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that : $f(f(x))=x^2-2$ to solve it i have no idea! can we solve it ...
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### Solving for best fit value $C$ in $\sqrt {\mathrm{Exp}_a^{[1/2]} (x) \cdot \mathrm{Exp}_b^{[1/2]} (x )} \sim\sim \mathrm{Exp}_C^{[1/2]} (x).$

Let $\mathrm{Exp}_t^{[y]} (x)$ denote the $y$ th iteration of the exponential function with base $t$ : $t^x.$ For example $\mathrm{Exp}_t^{[1]} (x) = t^x$. Let $\sim\sim$ denote best fit. Now as $x$...
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### Boundary of $x^2+x$ Julia set

How do you calculate an infinite fixed point for $f=x^2+x$, so that $f^{o n}(x)$ never repeats, and doesn't go to infinity and doesn't go to zero? Can such a sequence of points densely cover the ...
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let $f(x)=x^2$, then $f(f(x))=x^4$, so $x^4$ is a continuous function from $\Bbb R$ to $\Bbb R$ which can be obtained as $f\circ f$ for a continuous $f\colon \Bbb R\to \Bbb R$. general example: for $f(... 1answer 89 views ### Easy looking functional equation. [closed] Find all functions that satisfy $$f (f (x))=x^2-3x+4$$ Any thoughts and approachs to find$f (x)$? 1answer 57 views ### Existance of a function satisfying given equation Examine the existence of a function$f:\mathbb R\to \mathbb R$such that:$\forall x \in \mathbb R\ :f(f(x))= x^2 +1$How to approach this type of problems? I study functional equation from a ... 1answer 86 views ### half composite function There is half derivative. Is there any definition of half composite function? Or is it possible to define half composite of a function? 1)$f^n=f \circ \cdots \circ f $($n$-times,$n \in \mathbb Z^{+...
The main idea is that if you have some well defined function $f$, is there a function $f^{\frac{1}{2}}$ such that $f^{\frac{1}{2}}\left(f^{\frac{1}{2}}\left(x\right)\right) = f\left(x\right)$. For ...
So I was playing around with function composition, and started wondering if there was a way to split up a function in to a repeated application of another function. Notation: Let $f^1 = f$ and \$f^n = ...