Elementary questions about functions, notation, properties, and operations such as function composition.

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3answers
95 views

How would you reverse this double-variable equation?

So I have an example function: $$f(a,b) = \frac 12 ((a+b)^2 + 3a + b)$$ It "encodes" two variables into one unique number. Assuming a and b are always positive, how would you write a function to ...
1
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3answers
82 views

Prove $f : A\rightarrow B, g: B\rightarrow C$ , and $g\circ f: A \overset{1-1}{\rightarrow}C$, then $f:A\overset{1-1}{\rightarrow}B$

Could anyone please explain how to approach this problem, I'm honestly having a hard time figuring out where to start the problem. I know that I have to show that $\forall x,y\in A$ , if $f(x)=f(y)$, ...
3
votes
1answer
124 views

Addition formulas for Jacobi amplitude function

Are there any known summation formulas for the Jacobi amplitude function? I need a formula like $\mathrm{am}(t+x)=\mathrm{am}(t) + f(x)$. I have plotted some graphs and it seems that $f(x)$ is ...
1
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1answer
34 views

A function question; inverses

If $f(g(x)) = x$ for all $x$ and $f$ and $g$ are continuous. Does it necessary follow that $g = f^{-1}$? Or do we need $g(f(x)) = x$ as well?
4
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2answers
622 views

History of Functions

I am interested in the history of functions. Why did Euler introduce them? When and why did they become central to mathematics? I know the second question has something to do with the famous ...
1
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3answers
212 views

How to prove this property of floor function?

$ \left\lfloor { - x} \right\rfloor = - \left\lfloor x \right\rfloor $ if $x \in \mathbb{Z}$ and $ \left\lfloor { - x}\right\rfloor = - \left\lfloor x \right\rfloor -1 $ otherwise This is an ...
6
votes
3answers
204 views

Unambiguous terminology for domains, ranges, sources and targets.

Given a correspondence $f : X \rightarrow Y$ (which may or may not be a function) I generally use the following terminology. $X$ is the source of $f$ $Y$ is the target $\{x \in X \mid \exists y \in ...
1
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1answer
69 views

Does $f'(x) \in o(g'(x))$ imply $f(x) \in o(g(x))$ for monotonically increasing $f$ and $g$?

The title says it all. This seems intuitively true to me, but I'm not sure how one would go about proving this. (I'm asking because I'm trying to show that $x^n \in o(x^{n+1})$ for all natural $n$, ...
1
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1answer
27 views

How would you explicitly define the number type of a function's parameters?

Say I make a function $f$ that takes a parameter $a$, but I want to make sure that $a$ can only be $\mathbb{N}$, no $\mathbb{Z}$ or $\mathbb{R}$ allowed (as an example), how would I write that in a ...
5
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5answers
308 views

Is it possible to pass functions into other functions in maths?

I wanna be frank with you guys and say my mathematical education was a bit... bleh, so I'm teaching myself a lot of stuff lately, a question that has come up for me: "Is it possible to pass functions ...
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0answers
29 views

Help Understanding Provided Solution

I struggled with this problem for awhile before finally giving in and looking at the solution: "Let $n > 1$ be an integer, $A = \mathbb{Z}/n$ the integers modulo $n$ and $G$ the set of maps $\tau ...
0
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1answer
1k views

Supremum and Infimum of functions

I have been given the following homework problem... struggling. Any help would be appreciated. With the following functions state; a) State if the function is monotone. 1 b) Decide if it is ...
1
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0answers
190 views

Graph for lower and upperside bound absolute value function

The relation to be represented in graph is as follows $$ y = k \text{ for } k-1 < |x| < k \text{ where } k\in \mathbb Z $$ Normally we plot the area where the relation holds good is where ...
2
votes
2answers
170 views

Prove that $f$ has finite number of roots

Let $f:[0,1]\to \mathbb{R}$ be a differentiable function. If there do not exist any $x\in[0,1]$ such that $f(x)=f'(x)=0$, prove that $f$ has only finite number of zeros in $[0,1]$. I'm not ...
2
votes
3answers
66 views

Find domain of $ \sin ^ {-1} [\log_2(\frac{x}{2})]$

Problem: Find domain of $ \sin ^ {-1} [\log_2(\frac{x}{2})]$ Solution: $\log_2(\frac{x}{2})$ is defined for $\frac{x}{2} > 0$ $\log_2(\frac{x}{2})$ is defined for $x > 0$ Also ...
0
votes
1answer
105 views

Which way of writing functions is the most correct?

In functional programming it's not uncommon to bind a closure/lambda/anonymous function to a value name, i.e. $$f = x \mapsto x^2 + 3$$ so I've been wondering which is more right to do in ...
0
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1answer
36 views

How to define a function that gives us the number of pentagons formed in between two or more hexagons?

I have been trying to make a general formula/function that helps in calculating the number of pentagons that may be formed using 2 or more hexagons. Like it is shown in the picture below: In Fig. 1 ...
3
votes
1answer
62 views

Designing very simple function

I don't have much mathematical background except for highschool and I'm struggling to design a very simple function. I need a function f(x, y) that for the ...
1
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3answers
220 views

Show that $f$ is continuous at 0.

EDIT: Fixed the limit. This is a question from Spivak's Calculus, Ch.6, ex. 3. Suppose that $f$ is a function satisfying $$|f(x)|\leq |x| \forall x$$ Show that $f$ is continuous at 0. (Notice ...
16
votes
4answers
948 views

About the addition formula $f(x+y) = f(x)g(y)+f(y)g(x)$

Consider the functional equation $$f(x+y) = f(x)g(y)+f(y)g(x)$$ valid for all complex $x,y$. The only solutions I know for this equation are $f(x)=0$, $f(x)=Cx$, $f(x)=C\sin(x)$ and $f(x)=C\sinh(x)$. ...
6
votes
3answers
284 views

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 ...
0
votes
2answers
279 views

A Well-Defined Bijection on An Equivalence Class

DATA: Let $f:X\rightarrow Y$ be a surjective function. Define a relation $\sim$ on $X$ by $$a\sim b~\iff~f(a)=f(b).$$ Let $S=X/{\sim}$, namely let $S$ be the set of equivalence classes of elements ...
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2answers
122 views

$h:\mathbb{R}_{/\sim}\rightarrow \mathbb{R}^2$: A Bijection from a Quotient Space to the Unit Circle (Geometrically Considered)

NOTE: This is not a duplicate. Define a relation $\sim$ on $\mathbb{R}$ as follows: for any $a,b \in \mathbb{R}$, $$a\sim b \iff a-b\in \mathbb{Z}.$$ Let $S=\mathbb{R}_{/\sim}$. That is, $S$ is ...
2
votes
4answers
221 views

What is the contrapositive of the definition of onto?

I feel that the problem I am working on will be easier to prove by contrapositive, namely instead of showing surjective I want to show the contrapositive of surjective.
0
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2answers
112 views

Revisted$_2$: Are doubling and squaring well-defined on $\mathbb{R}/\mathbb{Z}$?

Define a relation $\sim$ on $\mathbb{R}$ as follows: for any $a,b \in \mathbb{R}$, $$a\sim b \iff a-b\in \mathbb{Z}.$$ Let $S=\mathbb{R}/{\sim}$. That is, $S$ is the set of equivalence classes of ...
0
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3answers
70 views

An Equivalence Relation: Introspection into a Particular Well-Defined Quotient

DATA: Let $f:\mathbb{Z}\setminus \{0\}\rightarrow \mathbb{N}$ be a function defined by $$f(n) = \{k~:~n=2^km,~m\in \cal{O}\},$$ where $\cal{O}$ is the set of odd integers. Let ...
2
votes
1answer
107 views

(Revisited$_2$) Injectivity Relies on The Existence of an Onto Function Mapping Back to Its Preimage

QUEST: For any sets $X$ and $Y$, there exists an injective function $f:X\rightarrow Y$ if and only if there exists a surjective function $g:Y\rightarrow X$. QUESTION$_1$: How do you people ...
2
votes
2answers
82 views

Unique root to a function

Let $f:[a,\infty)\rightarrow \mathbb{R}, \ \ f\in C^2[a,\infty)$ such that $$ \\ f(a)>0 , \ \ f'(a)<0, \ \ f''(x)\leq 0 \ \ \forall x\in [a,\infty)$$ Prove that $$ \exists !~t\in ...
1
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1answer
57 views

Approximating a function

I'm sorry if this question in not well formed... I would like to perform a computation of the following function: f() = -2*X1 -1*X2 +0*X3 + 1*X4 +2*X5 (The ...
2
votes
1answer
46 views

I need an equation for some data points.

My data points are (97.57,6.14), (90.54,7.03), (81.99,8.55), (71.47,10.52), (56.5,14.97) and (31.88,24.62). I'm trying to find the nonlinear equation that describes these points, but I'm having ...
1
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3answers
432 views

Left Inverse: An Analysis on Injectivity

I'm told that $g$ is a left inverse of $f$ if $g\circ f=1_X$. I'm also told that if $f$ has a left inverse, then $f$ must be injective. I'm now asked to prove the converse, namely that if ...
1
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4answers
122 views

(Revisted) Invertibility is necessary and sufficient for bijectivity…

Let $f:X\rightarrow Y$ be non-surjective (not onto), and let $g:Y\rightarrow Z$ be non-injective (not $1-1$). Now, construct the composition $g\circ f$ such that it's a bijection. My approach was ...
1
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1answer
143 views

Must the composition of a function with a surjective function be surjective?

I'm looking here, and I don't see an example like this: $$\begin{eqnarray} \zeta_1 & \xrightarrow{f} \omega_1 & \xrightarrow{g} \epsilon_1 \\ \zeta_2 & \rightarrow \omega_2 & ...
1
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2answers
70 views

(Revisted) Surjectivity: Examples for Compositions

I'm asked to give an example where $f$ is surjective, but $g\circ f$ is not. I suspect that $f(x)=x$ and $g(x)=\frac{1}{x}$ will do the trick, namely for $f,g : \mathbb{R}\rightarrow \mathbb{R}$, ...
2
votes
2answers
77 views

Is this a vertical asymptote?

I have this function: $$ f(x) = (x+1) \cdot e^{\frac{1}{x}} $$ I have the two side limits: $$ \lim_{x \to 0^-} { (x+1) \cdot e^{\frac{1}{x}} } = 0 $$ $$ \lim_{x \to 0^+} { (x+1) \cdot ...
0
votes
2answers
96 views

Does there exist such a function? [closed]

Can we find a function $ f: \mathbb{R} \to \mathbb{R} $ that is continuous only at the points $ 1,2,\ldots,100 $?
1
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1answer
58 views

Need an easy CDF for Inverse transform sampling

I want to use inverse transform sampling to generate some random numbers, which all fall into a given interval $(0,x_{max})$. The numbers are not necessarily distributed evenly but can be "skewed". I ...
1
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1answer
95 views

Question on 1-1 & onto function

While I was studying relations & functions I came across with this question which I can't figure out the meaning of it. Please help. Let $X$ be the set of all strings of finite length consisting ...
1
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2answers
55 views

On the Definition of Surjectivity

So I'm told that for a function $f:X\rightarrow Y$ to be surjective then $$\forall y\in Y~,~\exists x\in X~,~f(x)=y,$$ so does this "$\exists$" imply more than one such $x$ can hit the same $y$? ...
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4answers
615 views

Difference between a function and a graph of a function?

Formally, I learned that a function $f: X \to Y$ is a subset $f \subset X \times Y$ subject to the condition that for every $x \in X$, there is exactly one $y \in Y$ such that $(x, y) \in f$. We write ...
3
votes
3answers
214 views

Example of an injective function $g$ and function $f$ such that $g\circ f$ is not injective

Give an example of a function $g$ which is injective, but for which its composition with $f$ is not, namely $g\circ f$. I suspect that $f(x)=0$ and $g(x)=x$ will do, am I right?
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1answer
75 views

Injectivity and Compositionality

Give an example of two functions $f$ and $g$ for which $f$ is injective, but $g\circ f$ is not. I suspect this will do: $$f(x) = x~\text{and}~g(x)=1$$ This is the image which leads me to this ...
2
votes
5answers
2k views

$y = -2\sin(x - \pi/3):\;$minimum & maximum values?

I have the following function $$y = -2\sin⁡(x-\pi/3),\quad 0\leq x\leq 2\pi$$ I know that $\sin x$ has max at $\pi/2$ and min at $3\pi/2$ but how would I use this information to find the solution to ...
1
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1answer
86 views

Distribution of the Inverse of a Random Variable

I am trying to figure out how to find the distribution of the inverse of a random variable. Say, $Y=X^{-1}$ where X can take negative values. The two ways I know to find the distribution of a random ...
3
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1answer
69 views

What is the domain of the following function?

Please tell me the domain of $y = \sin^{-1}(\sin(x))$ P.S. I think domain is $(-\infty, \infty)$ But my teacher says it is $(-\frac{\pi}{2}, \frac{\pi}{2})$. He says since $sine$ is a many one ...
1
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1answer
63 views

Is the source (and/or target) a group, or just its underlying set?

Consider the following statements. Let $G$ and $H$ denote groups and $f : G \rightarrow H$ denote an arbitrary function. Let $G$ and $H$ denote groups and $f : G \rightarrow H$ denote an arbitrary ...
0
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1answer
69 views

Fixed point and non-fixed point function

For constructing another proof I need two functions explicitly and therefore I was wondering whether there exists a function that has nowhere a fixed point and a function that (maybe depending on the ...
5
votes
2answers
563 views

Right Inverse for Surjective Function

Prove that if $f:X\to Y$ is a surjective function between sets, then there must exist a function $g:Y\rightarrow X$ such that $f\circ g=1_Y$. I know that the identity function is onto, and if $f$ ...
0
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1answer
78 views

Relation $R$: $R\circ R \subseteq R \implies R$ is transitive

Let $R$ be a relation on $X$, a set. If $R\circ R\subseteq R$, then is $R$ transitive?
8
votes
7answers
424 views

Why is the Fibonacci ratio though a decreasing function, it is alternating and decreasing?

I tried to find the ratio of consecutive terms of the Fibonacci series and found that it is a decreasing function and it converges . I tried it though a small code piece in python so that I can have a ...