# Tagged Questions

24 views

### At most n functions

Some background: I was trying to solve the functional equation f(f(x))=sin(x). I realized that $f(\pi n)$ is a root of f for all integers n, because $f(f(\pi n))=\sin(\pi n)=0$. Thus, we can write f ...
21 views

60 views

174 views

### if $f(mn)+f(m+n-1)=f(m)f(n)$How find $f(n)$

let $f:N^{+}\to Z$,and $f$ is monotonic nondecreasing,and such $$f(m)f(n)=f(mn)+f(m+n-1),f(4)=5$$ Find all $f(n)$ My try: let $$m=2,n=2\Longrightarrow f^2(2)=f(4)+f(3)$$ ...
320 views

### If $f:N→N$ such that $f(f(x))=3x$, then find $f(2013)$

If $f:N→N$ such that $f(f(x))=3x$, then what is the value of $f(2013)$?
75 views

### Is this a field of study?

Is there a name for an equation that takes the following form? $$F(f(x),f^{-1}(x),x)=0$$ A nice example being $$f(x)-f^{-1}(x)=0$$ because the solutions of this equation are their own inverses. ...
47 views

### Determining quadratic function of this word problem

I have this word problem in my homework: ...
63 views

### Solving functional equation 2

Problem: find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x^2+f(y))=(f(x))^2+y^4 +2f(xy),\ \ \ \forall x,y\in\mathbb{R}$$
43 views

### Solve a functional equation

Find all functions $f:[0,+\infty)\to [0,+\infty)$ such that $f(x)\geq \frac{3x}{4}$ and $$f(4f(x)-3x)=x,\forall x\in[0,+\infty)$$
29 views

### I tried this problem in the following way; is it right?

Let, f(x) be a twice differentiable function defined on (-1, 1) and f(0) = 1. Let, f(x) ≥ 0, f'(x) ≤ 0 and f''(x) ≤ f(x) for all x ≥ 0. Show that, f'(0) ≥ -√2. I am telling you what I did. First, ...
110 views

### Show that $f(x) = x$ if $f(f(f(x))) = x$.

If $f: \mathbb{R} \to \mathbb{R}$ and $f$ is strictly increasing, show that $f(x) = x$ if $f(f(f(x))) = x$. So this compulsorily ESTABLISHES that $f(x) = x$ only, and no other solution. So, merely ...
57 views

### Another functional equation

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$\forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
44 views

### Functional equation with $f(1)$ integer

Here is a nice problem: Let $f:R\rightarrow R$ be a function, R is the set of real numbers, satisfying the following properties: $f(1)$ is an integer and $xf(y)+yf(x)=(f(x+y))^2-f(x^2)-f(y^2)$, for ...
### Simple Functional Equation $\frac{f(a)-f(b)}{a-b}\cdot(-a)+f(a)=-ab$
Compute all real-valued functions $f$ so that the line between any two points on the graph $f$ intersects the $x$-axis at the product of those two points' $x$-coordinates times $-1$. (if we do some ...