Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
34,645 questions
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What is an applicable way of averaging an everywhere surjective function whose graph has Hausdorff dimension $2$ with zero $2$-d Hausdorff measure?
Suppose $f:\mathbb{R}\to\mathbb{R}$ is an explicit everywhere surjective function whose graph has Hausdorff dimension $2$ with a zero $2$-d Hausdorff measure.
Since the integral of $f$ w.r.t. $2$-d ...
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What notation should I use for domain [closed]
I’ve got an inequality
$$3x^2+2x+\frac{1}{3}>0$$
and the answer given in the book is
$$x\in\left(-\infty;-\frac{1}{3}\right)\cup\left(-\frac{1}{3};+\infty\right)$$
but I don’t really like this ...
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Determine all possible values for C such that cos(A)cos(B) + sin(A)sin(B)sin(C)=1 [closed]
I did: sin(A)sin(B)sin(C)=1-cos(A)cos(B)
then:
sin(c)=(1-cos(A)cos(B))/(sin(A)sin(B))
now I am completely stuck on it.
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Let $a>1$, $b>0$ and $c=a+b$. Solve in $\mathbb{R}$ the equation $ (a^{x}+b)^{\log_{c} a} = c^{x} - b$
The Problem
Let $a>1$, $b>0$ and $c=a+b$. Solve in $\mathbb{R}$ the equation
$$
(a^{x}+b)^{\log_{c} a} = c^{x} - b.
$$
My Idea
Let $k=\log_{c} a$, so that $0<k<1$ and $a=c^{k}$.
Introduce ...
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Does $|A| = |B| = |S|$, where $S\subseteq A\times B$ imply that $S$ is a bijection?
Suppose there's a set of ordered pairs/tuples $S \subseteq \{(a,b) \mid a \in A, b \in B\} = A \times B$, e.g. $S = \{(👍, 1), (🙂, 2), (🎉, 3), ...\}$.
Does $|A| = |B| = |S|$ imply that $S$ is a ...
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Principle of Recursive Definition (Royden 3rd ed. 1988)
I am trying to flesh out a formal proof for the Principle of Recursive Definition as stated in Royden, 3rd edition.
Principle of Recursive Definition: Let f be a function from a set $X$ to itself, ...
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USAMTS Inspired Function Problem
This problem is from the most recent USAMTS Round 2, which has ended.
Let $\Bbb{Z}^+$ denote the set of positive integers. Determine, with proof, whether there exist functions $f,g:\Bbb{Z}^+\to\Bbb{Z}...
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Seeking different ways you can find $x=8$ in $(x-6)^3 = x^{\frac{1}{3}}+6$
I thought about the problem of finding an x such that
$$
(x-6)^3 = x^{1/3} + 6
$$
for a secondary-school class, in a context where students were studying functions and their inverses. They eventually ...
- 5,192
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2
answers
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Missing function in composition of two functions
Let $(f\circ g)(x) =x^4+2x^3-3x^2-4x+6$ and $g(x)=x^2+x-1$. Find $f(x)$, it seem to be $f$ will have the formula $f(x)=ax^2+bx+c$. Plugging $g(x)$ in $f(x)$, we get $$ f(x^2+x-1)=a(x^2+x-1)^2+b(x^2+x-...
- 3,282
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why the domain is equal or more zero for y=1+(x) ^(1/3) [closed]
Why Wolfram Alpha as the domain of $y=1+(x)^{\frac{1}{3}}$ gives $x\geq0$ and not $\mathbb{R}$?
We know that the function is well defined for all real numbers, but why Wolfram Alpha gives me only ...
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answer
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The formula to convert any values from certain range to different range if minimum and maximum are known
I'm looking for a formula that would work to elevate my students' grades. What I'm trying to say is when the minimum score gotten by my student is $0$ and the maximum is $42$, I want to convert them ...
- 2,557
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Possible arrangements for any n number of distinct cubes [closed]
This problem has been bouncing around in my head for years, and I can't seem to make progress. I'll give the rules.
Cubes are all uniform in size with an edge length of 1 unit.
Cubes are located ...
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Prove that the iterate $f^{n}$ is a constant function.
Let $A \subset \mathbb{R}$ be a finite set with $|A| = n$ and let $f : A \to A$ satisfy the strict contraction condition $|f(x) - f(y)| < |x - y|$ for all $x \neq y$ in $A$. Prove that $f$ is not ...
- 399
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4
answers
659
views
Complex logarithm base 1
Is a logarithm with base 1 defined in the field of complex numbers? I have not found any information about this. In real numbers, this is uncertain because $ \ln(1) = 0 $ and
$ \log_a(b)= \frac {\ln(...
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Solving the equation $(2^{x}-1)^2 = \log_{2}\!\big((1+\sqrt{x})^2\big)$ [closed]
the problem
$\text{Solve the equation} \qquad (2^{x}-1)^2 = \log_{2}\!\big((1+\sqrt{x})^2\big).$
My idea
Define
$$
f(x) = (2^{x}-1)^2 - \log_{2}\!\big((1+\sqrt{x})^2\big), \qquad x \ge 0.
$$
The ...
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