I'm looking for a function $f(x)$ so that $f_{inv}(x) = \frac{1}{f(x)}$. Does such a function exist and if so, how do I find it?
38843
Is there a function whose inverse is exactly the reciprocal of the function? That is $f^{-1} = \frac{1}{f}$.  We know that the inverse of a function is not necessarily equal to its reciprocal in general.
98812
I tried to use the definition and arrived this far:  $|f(x)-f(y)|=\left|\frac{x}{1+|x|}-\frac{y}{1+|y|}\right|=\frac{|x-y+x|y|-y|x||}{(1+|x|)(1+|y|)}\leq|x-y+x|y|-y|x||$.  Any suggestion for ending the proof?  I also tried to prove that $\frac{x}{1+x}$ is uniformly continuous on $[0,\infty[$ and that $\frac{x}{1-x}$ is uniformly continuous on $[-\infty,0[$, but I wonder if we can use the definition with the function $f(x)=\frac{x}{1+|x|}$ itself.
53126
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?
110733
$f$ is an entire function with $\operatorname{Im}f \geq 0$. Then which of the followings are true: 1) $f$ is constant. 2) $\operatorname{Re}f$ is constant. 3) $f = 0$. 4) $f'$ is a non-zero constant.  That (3) &amp; (4) are wrong can be shown by using $f(z) = i$. But I'm clueless about the remaining two options.
120385
Suppose we are given a function $f(x_{1}, x_{2})$. Does showing that $\frac{\partial f}{\partial x_{i}} = 0$ for $i = 1, 2$ imply that $f$ is a constant? Does this hold if we have $n$ variables instead?
17832
$f(x)=\frac{2}{2+4^x}$  find $f(\frac{1}{2014})+f(\frac{2}{2014})+.....+f(\frac{2013}{2014})$  Please guide me through it, the only step I know is probably to eliminate the denominator ps. Not a homework
103505
I was wondering if there are functions for which $$f'(x) &gt; f(x)$$ for all $x$. Only examples I could think of were $e^x - c$ and simply $- c$ in which $c &gt; 0$. Also, is there any significance in a function that is always less than its derivative?    Edit: Thank you very much for all the replies. It seems almost all functions that apply are exponential by nature...  Are there more examples like - 1/x?  Again are there any applications/physical manifestations of these functions? [for example an object with a velocity that is always greater than its position/acceleration  is always greater than its velocity]
16668
I wonder  whether there is a function $f\colon\Bbb R\to\Bbb R$ with the folowing characteristic?   for every two real numbers $\alpha,\beta,\alpha\lt\beta$,  $$\{f(x):x\in(\alpha,\beta)\}=\Bbb R$$   I can't  say such a function does not exist, neither can I construct a example  Thanks a lot!
66537
I've leart in Calculus 1 that the derivetive of $f(x)$ is:  $$\lim_{h\to0} \frac{f(x+h) - f(x)}{h}$$.  suppose $f(x,y)$ is a function with 2 variables,  does $$f'(x,y) = \lim_{h\to0} \frac{f(x+h, y+h) - f(x,y)}{h}$$ ?  If not , what is the derivetive of function with 2 variables?
30015
Suppose $x&gt;0$ and we have function $f(x)=\frac{\sin x}{x}$ how can we show $$\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$$ I need a hint to show this property .Thanks in advance . I tried for $n=1 ,2$  by finding maximum of $|f'| ,|f''|$ but I get stuck to show for $n$
92968
In general can one say that for a random variable X:  $E[\frac{1}{X}] = \frac{1}{E[X]}$ ?  I've worked out a few examples where this works but I'm not sure how widely this is useful...
88810
Can anyone find a function $f$ with an $x$ such that $f&#39;(x)=0$, $f(x)$ is either a relative max/min and $(x,f(x))$ is an inflection point? In other words, suppose you're using the second derivative test, see that $f&#39;&#39;(x)=0$ and then note that it's actually an inflection point. Can you conclude that $(x,f(x))$ is not a relative extremum?  It's easy to find an example where $f&#39;&#39;(x)=0$ and there is an extremum--take $f(x)=x^4$, but in that case, there isn't actually an inflection point at 0.  I'm certainly open to the possibility that an inflection point can never be an extremum, though it seems like something I ought to have discovered by now, were it true.
25530
I'm looking for a function $f$, whose third derivative is $f$ itself, while the first derivative isn't.  Is there any such function? Which one(s)? If not, how can we prove that there is none?  Notes:   $x\longmapsto c\cdot e^x, c \in R$ are the functions whose derivative is itself. $x\longmapsto \cosh(x)={e^x+e^{-x}\over 2}$ and $x\longmapsto \sinh(x)={e^x-e^{-x}\over 2}$ have their second derivatives equal to themselves. $x\longmapsto f(x)$, has its third derivative equal to itself. $x\longmapsto \cos(x)$ and $x\longmapsto \sin(x)$ have their fourth derivatives equal to themselves.
52940
How many continuous function $f(x)$ exist such that $$\int_{0}^{1}f(x)\big(1-f(x)\big)\mathrm dx = \frac{1}{4}\text?$$ I want to find the number of such functions but I don't know how to proceed. It would be of great help if someone can tell me which theorem to apply or provide with any other hint. Any help is highly appreciated.
26909
My question is simple.  Let $f(x)=x^3\sin(\frac{1}{x}), x\in \mathbb{R}$. Is this function uniformly continuous on $\mathbb{R}$?  I think it is not uniformly continuous when $x$ goes to $\infty$
118455
Find all continous functions $f\colon\mathbb{R}\rightarrow\mathbb{R}$ satisfying  $f(x)=f(x^2+ 1/4)$  What I've tried so far:    suppose that $f$ is one-one thus $x=x^2+1/4$ ... $x=1/2$ then $f(x)=f(1/2)$ is constant   That's it, I'm stuck here.
29793
Assmue that $f(x)\in C(0,+\infty)$ and for any $x&gt;0$ have $$\int_{x}^{x^2}f(t)dt=\int_{1}^{x}f(t)dt$$  Find this $f$  My try: we have $$2xf(x^2)-f(x)=f(x)\Longrightarrow xf(x^2)=f(x),\forall x&gt;0$$ and then I can't works