Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
2,754
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How to define a bijection between $(0,1)$ and $(0,1]$?
How to define a bijection between $(0,1)$ and $(0,1]$?
Or any other open and closed intervals?
If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that'...
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Overview of basic results about images and preimages
Are there some good overviews of basic facts about images and inverse images of sets under functions?
58
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6
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Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist
I've been trying to solve the following problem:
Suppose that $f$ and $f'$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to\...
- 1,387
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3
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Calculating the total number of surjective functions
It is quite easy to calculate the total number of functions from a set $X$ with $m$ elements to a set $Y$ with $n$ elements ($n^{m}$), and also the total number of injective functions ($n^{\underline{...
- 3,092
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No continuous function switches $\mathbb{Q}$ and the irrationals
Is there a way to prove the following result using connectedness?
Result:
Let $J=\mathbb{R} \setminus \mathbb{Q}$ denote the set of irrational numbers. There is no continuous map $f: \mathbb{R} \...
- 5,698
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3
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Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large \frac{1}{z}$ by definition discontinuous at $0$?
Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large{\frac{1}{z}}$ by definition discontinuous at $0$?
Personally I would say: "no". In my view a function can only ...
- 152k
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Surjectivity of $f:S\to S$ implies injectivity for finite $S$, and conversely
Let $S$ be a finite set. Let $f$ be a surjective function from $S$ to $S$.
How do I prove that it is injective?
- 15k
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Do harmonic numbers have a “closed-form” expression?
One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq ...
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Do we really need polynomials (In contrast to polynomial functions)?
In the following I'm going to call
a polynomial expression an element of a suitable algebraic structure (for example a ring, since it has an addition and a multiplication)
that has the form $a_{n}x^{...
- 5,255
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There exists an injection from $X$ to $Y$ if and only if there exists a surjection from $Y$ to $X$.
Theorem. Let $X$ and $Y$ be sets with $X$ nonempty. Then (P) there exists an injection $f:X\rightarrow Y$ if and only if (Q) there exists a surjection $g:Y\rightarrow X$.
For the P $\implies$ Q part, ...
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Inverse of $f(x)=\sin(x)+x$
What is the inverse of
$$f(x)=\sin(x)+x.$$
I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet.
What about
$$f(x)=\sin(a \cdot x)+x$$
where ...
- 119
36
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3
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When functions commute under composition
Today I was thinking about composition of functions. It has nice properties, its always associative, there is an identity, and if we restrict to bijective functions then we have an inverse.
But then ...
- 5,608
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9
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On the functional square root of $x^2+1$
There are some math quizzes like:
find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$
such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$
If such $\phi$ exists (it does in this example), $\phi$ can ...
- 141k
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Is there a bijective map from $(0,1)$ to $\mathbb{R}$?
I couldn't find a bijective map from $(0,1)$ to $\mathbb{R}$. Is there any example?
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8
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Is there a way to get trig functions without a calculator?
In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a ...
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Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$.
Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.
I do not understand how to go about completing this problem or even where to start.
- 1,557
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5
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Why are removable discontinuities even discontinuities at all?
If I have, for example, the function
$$f(x)=\frac{x^2+x-6}{x-2}$$
there will be a removable discontinuity at $x=2$, yes?
Why does this discontinuity exist at all if the function can be simplified to $...
- 1,431
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2
answers
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I need a better explanation of $(\epsilon,\delta)$-definition of limit
I am reading the $\epsilon$-$\delta$ definition of a limit here on Wikipedia.
It says that
$f(x)$ can be made as close as desired to $L$ by making the independent variable $x$ close enough, but ...
- 1,223
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4
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How do I prove that $\arccos(x) + \arccos(-x)=\pi$ when $x \in [-1,1]$? [closed]
Prove that $\arccos x + \arccos(-x) = \pi$ when $x \in [-1,1]$.
How do I prove this? Where should I begin and what should I consider?
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Is There a Natural Way to Extend Repeated Exponentiation Beyond Integers?
This question has been in my mind since high school.
We can get multiplication of natural numbers by repeated addition; equivalently, if we define $f$ recursively by $f(1)=m$ and $f(n+1)=f(n)+m$, ...
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Derivative of a function with respect to another function. [duplicate]
I want to calculate the derivative of a function with respect to, not a variable, but respect to another function. For example:
$$g(x)=2f(x)+x+\log[f(x)]$$
I want to compute $$\frac{\mathrm dg(x)}{\...
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7
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How to evaluate fractional tetrations?
Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
- 681
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Why is an empty function considered a function?
A function by definition is a set of ordered pairs, and also according the Kuratowski, an ordered pair $(x,y)$ is defined to be $$\{\{x\}, \{x,y\}\}.$$
Given $A\neq \varnothing$, and $\varnothing\...
- 1,009
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votes
4
answers
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When do two functions become equal?
When do two functions become equal?
I have stumbled over this definition of equality of functions in elementary real analysis.
Let $X$ and $Y$ be two sets. Let $f:X\rightarrow Y$ and $g:X\...
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X,Y are independent exponentially distributed then what is the distribution of X/(X+Y)
Been crushing my head with this exercise. I know how to get the distribution of a ratio of exponential variables and of the sum of them, but i can't piece everything together.
The exercise goes as ...
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How do I divide a function into even and odd sections?
While working on a proof showing that all functions limited to the domain of real numbers can be expressed as a sum of their odd and even components, I stumbled into a troublesome roadblock; namely, I ...
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2
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Prove that the only eigenvalue of a nilpotent operator is 0?
I need to prove that:
if a linear operator $\phi : V \rightarrow V$ on a vector space is nilpotent, then its only eigenvalue is $0$.
I know how to prove that this for a nilpotent matrix, but I'm ...
- 1,347
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3
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bijection between $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$ [duplicate]
I understand that both $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$ are of the same cardinality by the Shroeder-Bernstein theorem, meaning there exists at least one bijection between them. But I can'...
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Does $f(a,b)$ being directly proportional to $a$ and $b$ separately imply that $f(a,b)$ is directly proportional to $ab?$
For example, in physics, if $$\text{F} \propto m_1m_2$$ and $$\text{F} \propto \frac{1}{r^2},$$ then $$\text{F} \propto (m_1m_2)\left(\frac{1}{r^2}\right)= \frac{m_1m_2}{r^2}.$$
This property (...
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Count number of increasing functions, nondecreasing functions $f: \{1, 2, 3, \ldots, n\} \to \{1, 2, 3, \ldots, m\}$, with $m \geq n$.
I stumbled upon a question given like:
Let $m$ and $n$ be two integers such that $m \geq n \geq 1$.
Count the number of functions $$f: \{1, 2, · · · , n\} \to \{1, 2, · · · , m\}$$ of the following ...
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