Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
42 questions from the last 30 days
7
votes
2
answers
358
views
Find all the functions $f \colon \mathbf R \to \mathbf R$, such that $f(x+\frac{1}{y})+f(y+\frac{1}{x})=2f(xy)$ for all $x,y \in \mathbf R$.
the problem
Find all the functions $f \colon \mathbf R \to \mathbf R$, such that $$f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{x}\right)=2f(xy)$$ for all $x,y \in \mathbf R$.
my idea
Plugging in ...
- 539
3
votes
2
answers
417
views
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is bounded above and $f(xf(y))+yf(x)=xf(y)+f(xy)$
Find all functions $f : \mathbb R \to \mathbb R$ such that: $f (x f ( y ))+y f ( x) = x f ( y ) + f ( x y )$, $\forall\ x , y \in \mathbb{R}$
and
b) $\exists M \in \mathbb R$ such that $f(x)<M$ ...
- 105
7
votes
1
answer
337
views
For which real $\beta$ there exist concave(convex upwards) functions $f, g: (0;1) \to (0;+\infty)$ such that $\frac{f(x)}{g(x)}=(1+x)^\beta$?
For which real $\beta$ there exist strictly concave(convex upwards) functions $f, g: (0;1) \to (0;+\infty)$ such that $\frac{f(x)}{g(x)}=(1+x)^\beta$?
My attempt: if we don't require $f>0$ and $g&...
- 593
1
vote
1
answer
227
views
Does "the preimage of a closed interval is a finite union of closed intervals" imply $f:\mathbb{R}\to\mathbb{R}$ is continuous?
Suppose I have a function $f:\mathbb{R}\to\mathbb{R}$ with the property that for any closed interval, its preimage is a finite union of closed intervals. Can I conclude that $f$ is continuous, or do ...
- 7,992
2
votes
3
answers
183
views
How to solve $ \lim\limits_{x\to+\infty} \!\!\left(\! \frac{x^{2}+3}{3x^{2}+1}\! \right)^{\!x^{2}}\!\!\!=0\;?$
I know it can be solved with Squeeze's theorem, but I want to verify that this more conventional method might also be valid.
$$ \lim_{x\to+\infty} \left(\frac{1+\frac{3}{x^{2}}}{1+\frac{1}{3x^{2}}}\...
- 23
4
votes
1
answer
373
views
Is this construction of two periodic functions $f$ and $g$ such that $f(x)+g(x)=x$ with axiom of choice correct? What if without?
Example
Let
$$
\Bbb{Z}[\sqrt{2}]=\{m+n\sqrt{2}\mid m, n\in\Bbb{Z}\}
$$
Consider the quotient set $\Bbb{R}/\Bbb{Z}[\sqrt{2}]$, where the equivalence relation $x\sim y$ is defined as $x-y\in\Bbb{Z}[\...
- 1,849
-4
votes
4
answers
244
views
Find $ \lim_{x\to+\infty} \left( \frac{x^{2}+3}{3x^{2}+1} \right)^{x^{2}}=0 $ [closed]
Problem
$$ \lim_{x\to+\infty} \left( \frac{x^{2}+3}{3x^{2}+1} \right)^{x^{2}}=0 $$
My Work
$$ \lim_{x\to+\infty} \left(\frac{1+\frac{3}{x^{2}}}{1+\frac{1}{3x^{2}}}\cdot\frac{1}{3} \right)^{x^{2}\cdot\...
1
vote
2
answers
143
views
Domain and range of $f(x)=\frac{\sqrt{x-5}}{\sqrt{3-x}}$ — is the domain empty or not?
I’m a high-school student working on finding the domain and range of the following function
$$f(x)=\frac{\sqrt{x-5}}{\sqrt{3-x}}$$
My reasoning (straightforward conditions):
For the numerator to be ...
- 221
-2
votes
3
answers
116
views
Find $ \lim_{x\to\infty} (\frac{3+x^{2}}{4x^{2}-1})^{x^{2}}=0 $ [closed]
I tried but I obtein an indeterminacy. I think that I am ignoring any property of notable limits maybe
$$ \lim_{x\to\infty} (\frac{3+x^{2}}{4x^{2}-1})^{x^{2}}\to $$
$$ \lim_{x\to\infty} (1-\frac{3x^{2}...
3
votes
1
answer
143
views
Number of real roots of the n-th iteration of $f(x) = x^3 - 3x + 1$
Given, $$f(x) = x^3 - 3x + 1$$
I was solving a problem to find the number of distinct real roots of the composite function $f(f(x)) = 0$.
By analyzing the graph of $f(x)$, we can observe the local ...
- 964
4
votes
1
answer
140
views
Does there exist a function such that $f(x), f(x)+\sqrt{3}, \sqrt{2}-f(x), f(x)+x$ are irrational for all irrational $x$?
Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ such that $f(x), f(x)+\sqrt{3}, \sqrt{2}-f(x), f(x)+x$ are irrational for all irrational $x$?
My attempt: I couldn't come up with any good ...
- 593
-1
votes
3
answers
120
views
Find $ \lim_{x\to \infty} \left( \frac{x-4}{x+1} \right)^{x+3}=e^{-5} $
$$ \lim_{x\to \infty} \left( \frac{x-4}{x+1} \right)^{x+3}=e^{-5} $$
I know that I am not making any change in the expression, I am just re-expressing it
$$ \lim_{x\to \infty} \left( 1+\frac{-5}{x+1} \...
3
votes
3
answers
192
views
Finding a function for $\sin(x)\sec(y) = \sin(y) + \sec(x)$
While messing around on desmos, I discovered the function $$\sin(x)\sec(y)=\sin(y)+\sec(x)$$ which appears as a warped sinusoid glide-reflected to fill the plane (graph in Desmos).
Each of these ...
-1
votes
2
answers
119
views
Find $ \lim_{x\to \infty} \frac{x^{2}+bx+c}{x-n}=\infty $
$$ \lim_{x\to \infty} \frac{x^{2}+bx+c}{x-n}=\infty $$
recalling the trinomial of the form:
$$ x^{2}+bx+c=(x+n)(x+m) $$
for some $n$ and $m$ such that
$$ m+n=c $$ $$ m\cdot n=b $$
$$ \lim_{x\to \infty}...
1
vote
2
answers
136
views
Show that $f$ is strictly increasing on $\mathbb{R}_+.$
Problem statement:
Let $f:\mathbb{R} \to \mathbb{R}$ be defined by
$$
f(x) = a_1^x + a_2^x + \dots + a_n^x,
$$
where $n \in \mathbb{N}, \quad n \ge 3,$ and $a_1, a_2, \dots, a_n > 0,$ all of them ...
- 539