Functional equation

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that : $\forall x\in\mathbb{R}\space f(x)=f(x-1)^2-2$

#Algebra

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

$f(x)=2cos({2}^{x}a)$ where $a$ is an arbitrary parameter.

Siddharth Chakravarty - 3 months, 3 weeks ago

Thanks! I find new one from this : $f(x)=2\cdot\cosh\left(a\cdot2^{x}\right)$

Zakir Husain - 3 months, 3 weeks ago

$y=2$ starts out asymptotic to both of your functions, and I checked that a function based on $y=-1$ will kinda oscillate around it with an increasing amplitude for some time, after that I don’t know

Jason Gomez - 3 months, 3 weeks ago

I think there are a lot of pathological functions satisfying the FE

ChengYiin Ong - 3 months, 3 weeks ago

@Yajat Shamji @Aryan Sanghi @Mahdi Raza @Jeff Giff @David Stiff @Vinayak Srivastava @Akela Chana @Siddharth Chakravarty

Zakir Husain - 3 months, 3 weeks ago

Not all functions but at the moment, $f(x) = 2$ and $f(x) = -1$ are the boring constant functions which work

Jason Gomez - 3 months, 3 weeks ago

Piecewise functions involving $-\phi$ and $\frac{1}{\phi}$ can be made

Jason Gomez - 3 months, 3 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$