Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
2,756
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If $g \circ f$ is the identity function, then which of $f$ and $g$ is onto and which is one-to-one? [closed]
Say $f:X\rightarrow Y$ and $g:Y\rightarrow X$ are functions where $g\circ f:X\rightarrow X$ is the identity. Which of $f$ and $g$ is onto, and which is one-to-one?
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Is 'every exponential grows faster than every polynomial?' always true?
My algorithm textbook has a theorem that says
'For every $r > 1$ and every $d > 0$, we have $n^d = O(r^n)$.'
However, it does not provide proof.
Of course I know exponential grows faster ...
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3
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Homeomorphism(topological spaces) version of Cantor–Bernstein–Schroeder theorem
Let $A$ , $B$ be topological spaces such that there for some subset $D$ of $B$ there is a homeomorphism form $A$ to $D$ and for some subset $E$ of $A$ there is a homeomorphism form $B$ to $E$ ; then ...
- 8,387
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1
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Prove that functions map countable sets to countable sets
One of my homework problems asks to prove that if $f:X\to Y$ is a function and $A$ is a countable subset of $X$, then $f(A)$ is countable.
I believe I have a valid proof by partitioning the set A ...
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What is the domain of $x^x$ as a real valued function?
Consider the function $f(x) = x^x$.
Wolfram alpha tells me that this function's domain is $x : x>0$, $x \in \mathbb{R}$. I can't see why it cannot be defined for a number like $(-2)$. I mean $(-2)^...
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3
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Can an indefinite integral depend on $c$?
If we have an indefinite integral $g(x)+c$ of a function $f,$ can we treat $c$ (entirely independent of $x$) as being some changing value?
For example, if we want to use the parameter $c$ to classify ...
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Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $ [closed]
Let $f\left( x\right)$ be a $C^{2}$ function on $\mathbb{R}$. Show that $$\sup \left| f'\left( x\right) \right| ^{2}\leqslant4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right|...
- 577
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6
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Create unique number from 2 numbers
is there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 and 30, 1 and 15, 4 and 60. In general, if I take 2 random numbers result must ...
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Notation for repeated application of function
If I have the function $f(x)$ and I want to apply it $n$ times, what is the notation to use?
For example, would $f(f(x))$ be $f_2(x)$, $f^2(x)$, or anything less cumbersome than $f(f(x))$? This is ...
- 749
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How do you define functions for non-mathematicians?
I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ...
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9
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Can there be a function that's even and odd at the same time?
I woke up this morning and had this question in mind. Just curious if such function can exist.
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In Group theory proofs what is meant by "well defined"
What is exactly meant or required for a mapping to be well defined? I was reading the First Isomorphism theorem (link), and the first thing the proof does is define a map and find out if it's well ...
- 1,161
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can any continuous function be represented as a sum of convex and concave function?
I read that any continuous function can be represented as a sum of convex and concave function, meaning for all $f(x)$, $f(x) = g(x) + h(x)$ where $g$ is convex and $h$ is concave.
There could be ...
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How to prove: Moment Generating Function Uniqueness Theorem
Many results are based on the fact of the Moment Generating Function (MGF) Uniqueness Theorem, that says:
If $X$ and $Y$ are two random variables and equality holds for their MGF's: $m_X(t) = m_Y(...
- 5,653
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Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples?
- 1,021
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How is the Integral of $\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx$
Can Some one tell me what this method is called and how it works With a detailed proof
$$\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx$$
I've been using this a lot in definite integration but haven't seemed ...
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Difference of mapsto and right arrow
Could someone please explain to me what is the difference in the two arrows$$\rightarrow$$ and $$\mapsto$$ For example in Probability wih Martingales (Willams)
Thank you.
- 3,848
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Why are linear functions linear?
I always thought linear functions need to satisfy
$$f(x+y)=f(x)+f(y).$$
I am a tad confused now, consider $f(x)=2x+3$. $f(1)=5$, $f(2)=7$, $f(1+2)=f(3)=9 \neq f(1)+f(2)$ which was what I thought ...
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If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?
Is is true that if a function is Riemann integrable, then it is Lebesgue integrable with the same value? If it's true, how to prove it? If it's false, what is a counterexample?
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Why use the derivative and not the symmetric derivative?
The symmetric derivative is always equal to the regular derivative when it exists, and still isn't defined for jump discontinuities. From what I can tell the only differences are that a symmetric ...
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The origin of the function $f(x)$ notation
What are the historical origins of the $f(x)$ notation used for functions? That is when did people start to use this notation instead of just thinking in terms of two different variables one being ...
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How to prove that there exist no functions $f,g:\Bbb{R}\to\Bbb{R}$ such that $f(g(x))=x^{2018}$ and $g(f(x))=x^{2019}$? [duplicate]
I have tried a little bit to solve the problem which goes as follows:
My intuition says that there exist no $f:\Bbb R\to\Bbb R$ such that $$f(g(x))=x^{2018}\text{ and }g(f(x))=x^{2019}.$$
Note that ...
- 3,144
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What is the right way to define a function?
Most authors define functions this way:
Given the sets $A$ and $B$. A relation is a subset of $A\times B$. Then given a relation $R$, we define $Dom_R=\{x|(x,y)\in R\}$ and $Img_R=\{x|(y,x)\in R\}$.
...
14
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Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$
Suppose that $f(x)$ is differentiable on $[0,1]$ and $f(0) = f(1) = 0$. It is also
known that $|f''(x)| \le A$ for every $x \in (0,1)$. Prove that
$|f'(x)| \le A/2$ for every $x \in [0,1]$.
I'll ...
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If every point is a local maximum, is it a step function?
What are the functions $f:\mathbb R\to\mathbb R$ such that every point is a local maximum?
Certainly, $f(x)=c$ works for every constant. So does $\lfloor x\rfloor$, as does $I_{\{0\}}(x)=\begin{cases}...
- 23.5k
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Example where $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective
Can anyone come up with an explicit example of two functions $f$ and $g$ such that: $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective?
I tried the following: $$f:\mathbb{R}\rightarrow \...
- 949
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1
answer
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Function in $H^1$, but not continuous
By the Sobolev embedding theorem, if $\Omega$ is bounded, $H^s(\Omega)\subset C(\Omega)$ for $s>1$, in $\mathbb R^2$. Where I can find a counterexample (if one exists) for the case $s=1$? I mean a ...
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Proof that the Dirichlet function is discontinuous
I think I don't understand how it works.. I found some proofs.. okay, let's see:
Well I'd like to show that the function,
$$f(x) = \begin{cases} 0 & x \not\in \mathbb{Q}\\ 1 & x \in \mathbb{...
- 2,291
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Are all multiplicative functions additive?
Suppose $cf(x)=f(cx)$ and $f:\mathbb{R}\to\mathbb{R}$. I believe it follows that $f(x+y)=f(x)+f(y)$.
Proof: There is some $c$ such that $y=cx$. Then
$$f(x+y)=f\left((1+c)x\right)=(1+c)f(x)=f(x)+cf(...
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If $|A|=30$ and $|B|=20$, find the number of surjective functions $f:A \to B$.
Let there be: $|A|=n$ and $|B|=m$ if $m>n$ then there are $$m(m-1)\cdots(m-n+1)$$ injective functions, so in this case we have $|A|=30$ and $|B|=20$ that means $m<n$ so there exists a ...
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