# Functional Equation

#### idontknow

How to solve the equation ?
$$\displaystyle f(x)=\sqrt{f(x^2 )}$$ .

#### Integrator

How to solve the equation ?
$$\displaystyle f(x)=\sqrt{f(x^2 )}$$ .
Hello,

1) If $$\displaystyle f(x)\in \mathbb R$$ , then we get the equation $$\displaystyle f(x)=|f(x)|$$ and so we get an identity $$\displaystyle f(x)=f(x)$$ for $$\displaystyle f(x)\geq 0$$ and $$\displaystyle f(x)=0$$ for $$\displaystyle f(x)<0$$.
2) If $$\displaystyle f(x)\in \mathbb C$$ with $$\displaystyle Re(f(x))\neq 0)$$ and $$\displaystyle Im(f(x))\neq 0$$ , then we get the equation $$\displaystyle f(x)=f(x)\cdot (-1)^{\bigg[\frac{1}{2}-arctg \bigg(\frac{Im(f(x)}{\pi\cdot Re(f(x))}\bigg)\bigg]}$$ because $$\displaystyle -1=e^{i\pi}$$ where $$\displaystyle i^2=-1$$ and $$\displaystyle \bigg[\frac{1}{2}-arctg \bigg(\frac{Im(f(x)}{\pi\cdot Re(f(x))}\bigg)\bigg]$$ is the integer part of $$\displaystyle \frac{1}{2}-arctg \bigg(\frac{Im(f(x)}{\pi\cdot Re(f(x))}\bigg)$$.

All the best,

Integrator

P.S.
Thousands of apologies!The reasoning above is for $$\displaystyle f(x)=\sqrt{(f(x))^2}$$ and no for $$\displaystyle f(x)=\sqrt{f(x^2 )}$$.Thousands of apologies!

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#### skipjack

Forum Staff
1) If $$\displaystyle f(x)\in \mathbb R$$, then we get the equation $$\displaystyle f(x)=|f(x)|$$
Why? That doesn't seem to be implied.

1 person

#### Integrator

Why? That doesn't seem to be implied.
Hello,

Thousands of apologies! The reasoning above is for $$\displaystyle f(x)=\sqrt{(f(x))^2}$$ and no for $$\displaystyle f(x)=\sqrt{f(x^2)}$$. Thousands of apologies! Thank you very much for the correction!

All the best,

Integrator

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1 person

#### Integrator

How to solve the equation ?
$$\displaystyle f(x)=\sqrt{f(x^2 )}$$ .
Hello,

Did not You want to write $$\displaystyle f(x)=\sqrt{(f(x))^2}$$?If not, then it depends on the expression of the function $$\displaystyle f(x)$$.For exemple , if
$$\displaystyle f(x)=2x+3$$ , then the equation $$\displaystyle 2x+3=\sqrt{2x^2+3}$$ has the solution $$\displaystyle x=\sqrt6-3$$.
All the best,

Integrator

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#### DarnItJimImAnEngineer

I think the problem is to find the set of functions $$\displaystyle f(x)$$ such that the equation is satisfied either $$\displaystyle \forall x \in \mathbb{R}$$ or $$\displaystyle \forall x \in \mathbb{C}$$. Correct?

#### skipjack

Forum Staff
Let's assume that the function is from $\mathbb{R}$ to $\mathbb{R}$.

As $f(0)$ is 0 or 1, one can combine either of those possibilities with a definition of $f(x)$ for other values of $x$. For example, $f(x) = 0$ or $f(x) = |x|^r$, where $r$ is a real number.

2 people

#### Integrator

Let's assume that the function is from $\mathbb{R}$ to $\mathbb{R}$.

As $f(0)$ is 0 or 1, one can combine either of those possibilities with a definition of $f(x)$ for other values of $x$. For example, $f(x) = 0$ or $f(x) = |x|^r$, where $r$ is a real number.
Hello,

Correct, but if You want to solve the proposed equation as a functional equation, then I think that $$\displaystyle f(x)=|x|^c$$ is valid $$\displaystyle \forall x,c \in \mathbb C$$ with except $$\displaystyle x=0$$ and $$\displaystyle c=0$$ at the same time or $$\displaystyle x\in \bigg[ -\frac{1}{5},0 \bigg)\cup \bigg(0,+\frac{1}{5} \bigg]$$. Correct? Thank you very much!

All the best,

Integrator

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1 person