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
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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(...
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Jensen inequality for concave functions
Suppose $f(x)=\sqrt x$. From Jensen Inequality, we know that $f(x_1)+f(x_2)-2f((x_1+x_2)/2)\leq0$.
I am trying to show that if $x_3>x_1$, $x_3>x_2$, $x_4>x_1$ and $x_4>x_2$, then
$$
f(x_1)+...
- 52.5k
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Possible arrangements for any n number of distinct cubes
This problem has been bouncing around in my head for years, and I can't seem to make progress. I'll give the rules. Once I get a handle on
Cubes are all uniform in size with an edge length of 1 unit.
...
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Prove that the iterate $f^{n}$ is a constant function.
Let $A \subset \mathbb{R}$ be a finite set with $|A| = n$ and let $f : A \to A$ satisfy the strict contraction condition $|f(x) - f(y)| < |x - y|$ for all $x \neq y$ in $A$. Prove that $f$ is not ...
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Maximisation of functions of the form $f(x) = \sqrt{1 - x^2} + (ax+b)x$
I am studying a function arising in the analysis of robust aggregation rules in distributed learning, but the question is purely analytical. The function I am facing depends on parameters $a, b > 0$...
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Does "the preimage of a closed interval is a finite union of closed intervals" imply $f:\mathbb{R}\to\mathbb{R}$ is continuous?
Suppose I have a function $f:\mathbb{R}\to\mathbb{R}$ with the property that for any closed interval, its preimage is a finite union of closed intervals. Can I conclude that $f$ is continuous, or do ...
- 7,992
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Solving the equation $(2^{x}-1)^2 = \log_{2}\!\big((1+\sqrt{x})^2\big)$ [closed]
the problem
$\text{Solve the equation} \qquad (2^{x}-1)^2 = \log_{2}\!\big((1+\sqrt{x})^2\big).$
My idea
Define
$$
f(x) = (2^{x}-1)^2 - \log_{2}\!\big((1+\sqrt{x})^2\big), \qquad x \ge 0.
$$
The ...
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How to solve $ \lim\limits_{x\to+\infty} \!\!\left(\! \frac{x^{2}+3}{3x^{2}+1}\! \right)^{\!x^{2}}\!\!\!=0\;?$
I know it can be solved with Squeeze's theorem, but I want to verify that this more conventional method might also be valid.
$$ \lim_{x\to+\infty} \left(\frac{1+\frac{3}{x^{2}}}{1+\frac{1}{3x^{2}}}\...
- 19.5k
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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 ...
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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 ...
- 92.3k
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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 ...
- 3,478
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$f(x)$ is periodic with period p. [closed]
Suppose $f(x)$ is periodic with period p and $g(x)$ is periodic with period q. Let $r$ be the L.C.M. of p and q, if it exists. Then show that:
If $f(x)$ and $g(x)$ cannot be interchanged by adding a ...
CommunityBot
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Is there a combinatorial proof that the Catalan number $C_n$ satisfies $(n+1)C_n={2n \choose n}$?
I saw this question and thought that may be it is possible to prove that the $n^{\text{th}}$ Catalan number $C_n$ equals $\frac{1}{n+1}{2n\choose n}$ by taking a set $A$ of size $n+1$ and another set $...
- 564
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Remote generation of functions
The following problem appeared in my current quest for understanding fundamental physics. It is a bit complicated, but I try to explain it as clearly as possible. The problem has to do with the ...
- 216
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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 ...
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