Question

Proof that $\frac{3-x^{4}}{8x^{2}}$ is the only solution to the functional equation $f(x) + 3f\left(\frac{1}{x}\right) = x^{2}$

score 92 · Answer 1 · 2020-11-04 11:22:12

Proof that $\frac{3-x^{4}}{8x^{2}}$ is the only solution to the functional equation $f(x) + 3f\left(\frac{1}{x}\right) = x^{2}$

score 56 · Answer 2 · 2026-03-27 16:22:55

56

Views

Show that for all $z \in \mathbb{C}$, $f(z)=zf(1)$, if $f$ has the additive property.

Published on 27 Mar 2026 - 16:22

score 53 · Answer 3 · 2026-04-12 19:17:57

53

Views

Expansion of $\cosh(\sqrt{1+x^2})$ in the basis of $\cosh(\lambda x)$

Published on 12 Apr 2026 - 19:17

score 119 · Answer 4 · 2026-04-12 15:27:22

119

Views

Solutions to the functional equation $f(f(x))=f(x+1)$

Published on 12 Apr 2026 - 15:27

score 87 · Answer 5 · 2026-03-28 16:26:30

87

Views

Functions which satisfy $f(m^2+n^2)=f(m)^2+f(n)^2$

Published on 28 Mar 2026 - 16:26

score 493 · Answer 6 · 2020-11-10 10:45:49

493

Views

Find $f$ so that $f(f(n))=f(f(n+2)+2)=n$

Published on 10 Nov 2020 - 10:45

score 64 · Answer 7 · 2026-04-12 02:24:12

64

Views

nonexistence of a function satisfying $f(x)^{f(y)}=y^x$

Published on 12 Apr 2026 - 2:24

score 73 · Answer 8 · 2026-04-12 02:54:24

73

Views

Are there other solutions to the functional equation $f(x,y)\circ f(y,x)=e$?

Published on 12 Apr 2026 - 2:54

score 43 · Answer 9 · 2020-11-11 05:39:59

43

Views

If $\varphi$ is continuous, find a continuous branch of $t\mapsto\ln(\varphi(t))$

Published on 11 Nov 2020 - 5:39

score 91 · Answer 10 · 2020-11-11 18:09:14

91

Views

If $f(x)=x^2+ax+b$ has integer roots, and $f(x+\frac1x)=f(x)+f(\frac1x)$, then find $a^2+b^2$

Published on 11 Nov 2020 - 18:09

score 50 · Answer 11 · 2026-04-12 08:10:07

50

Views

Solving this functional equation involving fixed polynomials

Published on 12 Apr 2026 - 8:10

score 50 · Answer 12 · 2026-04-03 19:42:25

50

Views

Are there any functions that satisfy these properties?

Published on 03 Apr 2026 - 19:42

score 103 · Answer 13 · 2026-04-12 21:02:02

103

Views

Two functions with a given difference quotient composition (motivated by the MVT)

Published on 12 Apr 2026 - 21:02

score 67 · Answer 14 · 2020-11-15 04:48:17

67

Views

Solve for $f$ in an ODE involving two points: $f'(x)-f'(y)\geq g(x,y)$ where $g$ is a known function.

Published on 15 Nov 2020 - 4:48

score 454 · Answer 15 · 2020-11-15 06:14:55

454

Views

$f(yf(x) + y) = xy + f(y)$ for all $x, y.$ Prove $f$ is surjective

Published on 15 Nov 2020 - 6:14

List Question

Proof that $\frac{3-x^{4}}{8x^{2}}$ is the only solution to the functional equation $f(x) + 3f\left(\frac{1}{x}\right) = x^{2}$

Show that for all $z \in \mathbb{C}$, $f(z)=zf(1)$, if $f$ has the additive property.

Expansion of $\cosh(\sqrt{1+x^2})$ in the basis of $\cosh(\lambda x)$

Solutions to the functional equation $f(f(x))=f(x+1)$

Functions which satisfy $f(m^2+n^2)=f(m)^2+f(n)^2$

Find $f$ so that $f(f(n))=f(f(n+2)+2)=n$

nonexistence of a function satisfying $f(x)^{f(y)}=y^x$

Are there other solutions to the functional equation $f(x,y)\circ f(y,x)=e$?

If $\varphi$ is continuous, find a continuous branch of $t\mapsto\ln(\varphi(t))$

If $f(x)=x^2+ax+b$ has integer roots, and $f(x+\frac1x)=f(x)+f(\frac1x)$, then find $a^2+b^2$

Solving this functional equation involving fixed polynomials

Are there any functions that satisfy these properties?

Two functions with a given difference quotient composition (motivated by the MVT)

Solve for $f$ in an ODE involving two points: $f'(x)-f'(y)\geq g(x,y)$ where $g$ is a known function.

$f(yf(x) + y) = xy + f(y)$ for all $x, y.$ Prove $f$ is surjective

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