Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
34,641 questions
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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.
Cubes are all uniform in size with an edge length of 1 unit.
Cubes are located ...
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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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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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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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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 ...
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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 ...
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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.
$$ \lim_{x\to+\infty} \left(\frac{1+\frac{3}{x^{2}}}{1+\frac{1}{3x^{2}}}\...
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Number of real roots of the n-th iteration of $f(x) = x^3 - 3x + 1$
Given, $$f(x) = x^3 - 3x + 1$$
I was solving a problem to find the number of distinct real roots of the composite function $f(f(x)) = 0$.
By analyzing the graph of $f(x)$, we can observe the local ...
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What is the relationship between the two different definitions of Concave Function? [duplicate]
Some articles indicate the definition of a concave function $f(x)$ as follows:
$$\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
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Proof of average rate of change of a function
I was learning about derivatives and I saw that the slope of the secant line between two points is the average rate of change of the function between the two points. But, the average, as we normally ...
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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 ...
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How to prove that $C^k(\mathbb{R})$ is a subspace of $F(\mathbb{R};\mathbb{R})$?
I'm starting my linear algebra studies and came across the following statemtent:
$E = F(\mathbb{R};\mathbb{R})$ is the vector space of the one variable real functions $f:\mathbb{R} \rightarrow \...
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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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Expressing function in terms of preferred variables
Suppose we have $2$ vector valued functions of time,$\vec R(t)$ and $\vec r(t)$. We can represent those functions as:-
$$
\begin{split}
\vec R(t)&=\sum R_i(t)\hat I \\
\vec r(t)&=\sum r_i(t)\...
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