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

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1answer
33 views

If $\int_{0}^{\pi}t^{n}f(t)dt=0$ for all $n\in\{0\}\cup \mathbb N$ then prove that $f \equiv0$

Let, $f:[0,\pi ]\to \mathbb R$ be a continuous function such that $f(0)=0$. If $\int_{0}^{\pi}t^{n}f(t)dt=0$ for all $n\in\{0\}\cup \mathbb N$ then prove that $f \equiv0$. Since, $f$ is ...
3
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5answers
56 views

Example $g\circ f=id_A$ but $f\circ g\neq id_B$

Two functions $f:A\rightarrow B$ and $g:B\rightarrow A$. Can someone give me an example where $g\circ f=id_A$ but $f\circ g\neq id_B$?
0
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2answers
37 views

Is there a non-ambiguous name for the “square of a function”?

Given a function $f$, I want to refer to $f \circ f$ other than by a formula. Is there any name for this other than square of $f$, which has the problem of being ambiguous? In analogy to the ...
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0answers
28 views

Trying to extrapolate function (some kind of $x^3$ shape)

I would need to get a generic function $f(x)$, with variable $m < z < M$ from this characteristics: $f(m) = 0$ $f(z) = 50$ $f(M) = 100$ $f'(z) = 0$ $f'$ is a increasing function with $[m,M]$ ...
0
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1answer
38 views

Find range of values for function to be onto

Suppose $f:\mathbb R\rightarrow\mathbb R$ where $f(x)=\dfrac{ax^2+6x-8}{a+6x-8x^2}$ Find the range of values of $a$ for which $f$ is onto. I tried many thing like assuming it to be $y$ then the ...
0
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1answer
29 views

Small question: Name for the x of function f such that f(x)=x?

Background When doing maths and chemistry problems, I often came across things like $$x-\frac{x}{2}=\frac{x}{2}$$ It might seems trivial, but I found that it is often the presence of expressions like ...
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0answers
13 views

What is the domain and image of the composition of mappers in a manifold

I was trying to understand the following: which I got from: http://www.mit.edu/~9.520/fall14/slides/class14/class14_manifold.pdf I was wondering, why the domain was: $$ \alpha(U_{\alpha} \cap ...
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0answers
36 views

Interpretation of an integral of a function $f$

When we think of a Riemann integral, it is usually defined as $\lim_{\Delta x_{k}\rightarrow 0}\sum_{k = 1}^{n}f(x_{k}^{*})\Delta x_{k} = \int_{a}^{b}f(x)~dx$. This means that $f(x)$ should be ...
0
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1answer
47 views

Defining the differentiation operator

The differentiation operator is the function $\frac{\mathrm{d} }{\mathrm{d} x}: f \mapsto f'$. My question is, does the operator really take an entire function $f$ as an argument? For example, when ...
1
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1answer
34 views

Function as onto

I have a doubt that if a function is one-to-one then it will also be onto. If a function $f(x)$ is defined such that $f: \mathbb{R} \rightarrow \mathbb{R}$ then if the function is many to one then ...
4
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1answer
22 views

Showing that f,g are invertible if $A$ is a finite set and $f,g: A\to A$ such that $f\circ g$ is invertible

Let $A$ be a finite set and $f,g: A\to A$ such that $f\circ g$ is invertible. Prove f,g are invertible. Prove that if $A$ is an infinite set, it doesn't mean that f,g are invertible. ...
1
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3answers
111 views

Handling division by zero axiomatically

Suppose we define the multiplicative inverse function on real numbers as follows: $\forall{x \in \mathbb{R}}(x \neq 0 \implies x \times \frac{1}{x} = 1) $. Consider this truth table. \begin{array} ...
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0answers
26 views

Finding out if a function is invertible: $f,g:\mathbb N\to \mathbb N$, $g(x)=2x$ and $f$ with cases

Let $f,g:\mathbb N\to \mathbb N$ such that $g(x)=2x$ and $f(x)=\begin{cases}\frac x 2 &, x\in\mathbb N_{even}\\ x+9 &,x\in\mathbb N_{odd}\end{cases}$ ...
2
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1answer
50 views

Recursively enumerable sets are domain of partial recursive functions

My definition of recursively enumerable set is that it is the language recognized by some Turing machine. I want to show that this definition is equivalent to "a r.e. set is the domain of some ...
0
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1answer
44 views

How do I prove that the only possible function is $exp$?

Let´s say we have a differentiable function $f : \mathbb{R} -> \mathbb{R}$ with $f' = f$ and $f(0) = 1$ . How do I show that the only possible function for this to work $f = exp$ ? ...
1
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2answers
274 views

Number of one -to-one functions

Let $A = \{1, 2, 3, 4\}$ and $B = \{a, b, c, d, e\}$. what is the number of functions from $A$ to $B$ are either one-to-one or map the element $1$ to $c$? My answer is $166$, but I'm not really sure ...
0
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1answer
31 views

Can every function be a composite to itself and how to know if a composite between two functions is defined?

Can every function be a composite to itself? like we have $f:A\to B$ is $f \circ f$ always defined? Can we say that if $f$ is a injection/surjection/bijection then so is $f\circ f$? Also, how do ...
3
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4answers
39 views

Double branch $\sqrt x$ or square function turned 90°?

I have this idea for a graph but don't know what function could describe it better. The idea is something like the "squared" function turned $90$ degrees to the right, so that possible values for $x$ ...
4
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2answers
53 views

Math Contest Question with Polynomials

Prove that there does not exist a polynomial f(x) with integer coefficients for which f(2008) = 0 and f(2010) = 1867. This is a question from CMOQR (Qualifier for Canadian Math Olympiad , not the ...
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0answers
12 views

Is there a name/notation for coordinate-wise identical function?

Let's define $g: \mathbb{U}^n \rightarrow \mathbb{V}^n$ where $\mathbb{U}$ and $\mathbb{V}$ are arbitrary sets as $$g(u) = \left[f(u_1), f(u_2), \ldots, f(u_n) \right]^T$$ for some $f: \mathbb{U} ...
1
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1answer
41 views

Find the function continuous or discontinuous

$\sum_{n=1}^∞ $ $(x+2)^n \over n! + x^2$ , Interval = [1,2] Is this function continuous in that interval ? I tried but the factorials are troubles.
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4answers
50 views

Writing a piecewise function for $f(x) = \mid x+3\mid -\mid x-1\mid $

I am wanting to write a piecewise function for the following: $$f(x)= \mid x+3\mid -\mid x-1\mid $$ I know how to write piecewise functions for functions that have a single set of absolute ...
4
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1answer
31 views

Graphing a Piecewise Function

I graphed this function below. I want to make sure I am graphing piecewise functions such as this one correctly.
1
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1answer
83 views

Inverse function for $y=\lfloor x\rfloor+x$

Find the inverse function of the following function: $y=\lfloor x\rfloor +x$ I have tried writing down $x$ as $\lfloor x\rfloor +\{x\}$ but didn't get anywhere with that. A proper hint ...
1
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1answer
46 views

Exponential function given two points

I am trying to find an exponential function satisfying two points (having base "exp"). After some search, I couldn't find something relative (the most relevant was that ...
0
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2answers
49 views

Is $f(x) =|x| - 3$ even, odd, or neither?

$$f(x)=|x|-3$$ Is the function above odd, even or neither? I know that a function is even if $f(x) = f(-x)$: $$f(-x) = |(-x)| - 3$$ $$f(-x) = x-3$$ Does this mean that the function is even? ...
0
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1answer
16 views

How to resemble hyperbolic trigonometric functions (HTF) from normal trigonometric functions(NTF)??

There are many properties of HTF similliar but little different than NTF, Is there some pattern or rule that makes me, who only knows NTF, able to get HTF from direct resemblence to NTF?
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5answers
1k views

Why does notation for functions seem to be abused and ambiguous?

I really need to clear up a few things about function notation; I can't seem to grasp how to interpret it. As of right now, I know that a function is roughly a mapping between a set $X$ and a set $Y$, ...
1
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0answers
30 views

Name for a nowhere constant function?

Is there a pithy name for a function $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$ such that there is no non-degenerate interval $I \subseteq \mathbb{R}^n$ such that $f$ is constant on $I$ (by '$f$ is ...
3
votes
3answers
43 views

Multiple variables calculus: condition for $f$ to be continuous using curves

Prove $f:\mathbb{R}^n\to\mathbb{R}$ is continuous iff for every curve, $\gamma:[a,b]\to\mathbb{R}^n: f\circ \gamma :\mathbb{R}\to\mathbb{R}$ is continuous. $(\Rightarrow)$ is trivial. ...
7
votes
4answers
57 views

Proving a set is uncountable. [duplicate]

I need to prove that the set of all functions $\mathcal{F}:\mathbb{N}\rightarrow \left \{ 0,1 \right \}$ is uncountable. I'm not too sure at all how to do this. My initial idea was to try and show ...
1
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2answers
49 views

Is the function $f(x) = tan(x)$ odd, even, or neither?

Is the function $f(x) = \tan(x)$ odd, even, or neither? Here is what I have so far: I know the function is not even because $f(x) ≠ f(-x):$ $$f(-x) = \tan(-x)$$ $$\tan(-x) ≠ \tan(x)$$ Now I ...
2
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1answer
36 views

Prove that f(x,y) is not continuous for any a element of R

Given function $$ f(x,y) = \begin{cases} 3xy/(x^2 + y^2) & (x,y) \neq (0,0) \\ a & (x,y) = (0,0) \end{cases}$$ prove there exists no $a \in \Bbb R$ ...
0
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1answer
50 views

How to prove $(2^{-1/y}(1-x)+x)^{-y}$ is increasing in $y$, when $x,y \in (0,1)$.

As the title suggests, how to prove $(2^{-1/y}(1-x)+x)^{-y}$ is increasing in $y$ when $x,y \in (0,1)$?
0
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4answers
58 views

limit of the function $x \to 1$

Why $$\lim_{x\rightarrow1} \frac{ e^{ \frac{1}{\pi} \ln x } -1} {\frac{1}{\pi} \ln x }=1$$ I do not get it at all... I do not want explanation that would involve use of l'hospital rule
0
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2answers
38 views

Optimization problem of two variable

Find two numbers $a$ and $b$ with $a \leq b$ such that $\int_a^b (6-x-x^2)dx$ has the largest value.
2
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4answers
66 views

Find a path of a particle

I'm stuck with the following question: Let P be a particle at point $(1,2)$ on the surface $z=x^2y^2$. At $t=0$ the particle is left and moves freely. Find the path that the particle passes during ...
0
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1answer
18 views

Injectivity using complex cubics.

I've got this question and I've never been taught about injectivity with regards to complex numbers. If it were for real numbers then I would find the value of alpha such that the derivative never ...
0
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1answer
51 views

How to understand point functions

I am having trouble understanding the meaning of point functions. I know the mathematical definition but i don't think that i truly understand there true meaning. Point functions: Suppose $(X,d)$ is ...
2
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3answers
32 views

Help solving volume equation

I'm working on some ecological data that uses the following equation to calculate volume (V) of timber logs in cubic metres per hectare based on surveys along a transect. $$ V=\frac{\pi^2\Sigma ...
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1answer
32 views

Determine the range of f(x)=(sinx)/x

I am having trouble understanding the solution to this question. ''Determine the range of the following function: $f(x)$ = $(1$ $if$ $x=0)$ or (${\sin x\over x}$ if $x$$\neq$$0$) where the domain ...
1
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3answers
78 views

How to find the points of intersection? $ y = x^2\text{ and } y = x + 6$ [closed]

I am very confused as to how to solve this equation. How to find the points of intersection? $$ y = x^2\;\;\text{ and }\;\; y = x + 6$$
0
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1answer
28 views

Finding a non-piece wise function that gives us the $i$'th largest number.

A friend of mine was asked to find a non-piece wise function on four variables $i,a,b,c$ such that $f(i,a,b,c)$ is the $i$'th largest number among $\{a,b,c\}$. Using max and min or defining the ...
0
votes
2answers
36 views

mean value theorem question

I was trying to solve the following: Given: $0 < a < b$ and $n>1$, prove: $$na^{n-1}(b-a) < b^n-a^n < nb^{n-1}(b-a)$$ I managed to get this far using the mean value theorem: ...
0
votes
4answers
67 views

Trigonometric function problem

Given: $f(x)=2\arctan(x) +\arcsin(2x/(1+x^2))$ prove that for every $x \ge 1, f(x)=\pi.$ any idea how to approach this question? thanks
1
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1answer
49 views

a problem in The Mean-Value Theorem [duplicate]

Let $f(x)$ be defined and continuous on the interval $[a,b]$ and differentiable on $(a,b)$. Prove that there is at least one number $c$ such that $$\frac{af(a)-bf(b)}{a-b}=f(c)+cf'(c).$$
0
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4answers
32 views

Polynomial Function Equation

I am having some trouble with a problem. Thanks in advance to anyone who answers. Find a and b if $4x^4 + ax^3 + bx^2 + 6x + 1 = |P(x)|^2$ I have been staring at this problem for quite some time, ...
0
votes
5answers
60 views

Can you find the formula of this function?

I tried with the form y=ax+b and $$y=ax^2+bx+c$$ but these forms weren't appropriate for it.
0
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1answer
28 views

Help proving a theorem on uniform continuity in an open interval

I need help proving that given f:(a,b)→R that is uniformly continuous, it is possible to extend f to f:[a,b]→R that is continuous on the closed interval. thanks in advance!
3
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1answer
41 views

Calculated $f(2004)$ where the function $f:N \rightarrow Q^*$

It is considered that the function $f:N \rightarrow Q^*$ which has properties: $a) f(7) = 4;$ $b) f(8013) = 8015;$ $c) f(n+2).f(n) = 1+ f(n+1).$ Calculated $f(2004)$. All my attempts to find the ...