Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
34,642 questions
1
vote
1
answer
46
views
Possible arrangements for any n number of distinct cubes
This problem has been bouncing around in my head for years, and I can't seem to make progress. I'll give the rules. Once I get a handle on
Cubes are all uniform in size with an edge length of 1 unit.
...
- 11
1
vote
2
answers
149
views
Prove that the iterate $f^{n}$ is a constant function.
Let $A \subset \mathbb{R}$ be a finite set with $|A| = n$ and let $f : A \to A$ satisfy the strict contraction condition $|f(x) - f(y)| < |x - y|$ for all $x \neq y$ in $A$. Prove that $f$ is not ...
- 539
3
votes
3
answers
143
views
Complex logarithm base 1
Is a logarithm with base 1 defined in the field of complex numbers? I have not found any information about this. In real numbers, this is uncertain because $ \ln(1) = 0 $ and
$ \log_a(b)= \frac {\ln(...
- 97
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
0
votes
1
answer
82
views
Maximisation of functions of the form $f(x) = \sqrt{1 - x^2} + (ax+b)x$
I am studying a function arising in the analysis of robust aggregation rules in distributed learning, but the question is purely analytical. The function I am facing depends on parameters $a, b > 0$...
- 936
-1
votes
0
answers
129
views
Solving the equation $(2^{x}-1)^2 = \log_{2}\!\big((1+\sqrt{x})^2\big)$ [closed]
the problem
$\text{Solve the equation} \qquad (2^{x}-1)^2 = \log_{2}\!\big((1+\sqrt{x})^2\big).$
My idea
Define
$$
f(x) = (2^{x}-1)^2 - \log_{2}\!\big((1+\sqrt{x})^2\big), \qquad x \ge 0.
$$
The ...
- 539
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.
$$ \lim_{x\to+\infty} \left(\frac{1+\frac{3}{x^{2}}}{1+\frac{1}{3x^{2}}}\...
- 23
3
votes
1
answer
143
views
Number of real roots of the n-th iteration of $f(x) = x^3 - 3x + 1$
Given, $$f(x) = x^3 - 3x + 1$$
I was solving a problem to find the number of distinct real roots of the composite function $f(f(x)) = 0$.
By analyzing the graph of $f(x)$, we can observe the local ...
- 964
1
vote
1
answer
107
views
A question about the formal definition of a function graph.
With some friends I am currently reading and trying to understand Category Theory by Steve Awodey. As I am no trained mathematician, even simple issues can halt my progress. One occurred when I tried ...
- 21
3
votes
3
answers
192
views
Finding a function for $\sin(x)\sec(y) = \sin(y) + \sec(x)$
While messing around on desmos, I discovered the function $$\sin(x)\sec(y)=\sin(y)+\sec(x)$$ which appears as a warped sinusoid glide-reflected to fill the plane (graph in Desmos).
Each of these ...
2
votes
0
answers
25
views
What is the relationship between the two different definitions of Concave Function? [duplicate]
Some articles indicate the definition of a concave function $f(x)$ as follows:
$$\forall x_1,x_2\in D_f, \forall\lambda\in(0,1): f\left((1-\lambda)x_1+\lambda x_2\right) > (1-\lambda)f(x_1) + \...
- 23
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
1
answer
57
views
Differentiable functions
If we have a function $g(x)$ defined by $g(x) = f_1(x)f_2(x)$ where $f_1(x)$ and $f_2(x)$ are non-differentiable at some points, can $g(x)$ ever be differentiable everywhere? Intuitively using product ...
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
3
votes
1
answer
73
views
Range of base $a$ such that $f(x) = a^x - bx + e^2$ has two distinct zeros for all $b > 2e^2$
I am trying to solve the following problem involving a function with parameters $a$ and $b$.
The Problem:
Given the function $f(x) = a^x - bx + e^2$ where $a > 1$ and $x \in \mathbb{R}$.
Discuss ...
