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

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2answers
87 views

Finding a Solution to a Equation that Ends up as a Weird Repeating Series

I need to find the solution to this equation that ends up in a weird repeating series. The equation in question is: $$ \ln(y)=\frac{K}{\alpha}+\frac {x^{2}}{2\alpha\sigma}+\frac{\ln(\ln(y))}{2\alpha} $...
0
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4answers
116 views

When $f(0) = 0,$ is it always true that $G(0) = 0,$ where $G$ is the antiderivative of $f$?

I have a hunch that it is, but it would be nice if somebody could confirm / disprove it for me. Thank you. Edit Is it when the constant of integration is equal to zero?
2
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3answers
154 views

$\sin x + x\cos x=0$ solution?

Any idea of solving this equation? $$\sin x + x\cos x=0$$ I have also tried by setting a function $g(x)=\sin x+x\cos x$ and searching for solutions using the derivative but my atempts w
0
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3answers
75 views

Is $f(-x) = 1/f(x)$ under these conditions? [closed]

Let $f\colon \mathbb{R}\to \mathbb{R}$, so that: for $x\in \mathbb{R}$, $f(x)>0$ for $x,y \in \mathbb{R}$, $f(x+y) = f(x)f(y)$ Prove that $f(-x) = 1/f(x)$.
-1
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4answers
103 views

Calculate function: $\int_{a}^{b} \left(f{(x)}\right)dx=c$

Is there a way to find the function $f{(x)}$ for a given value of $a,b,c$? $$\int_{a}^{b} \left(f{(x)}\right)dx=c$$ For example: $a=0,b=1,c=\frac{1}{3}$ we get: $$\int_{0}^{1} \left(f{(x)}\right)...
0
votes
0answers
26 views

Inverse mappings and transition functions

Could somebody tell me if this is correct? I'm trying to understand mappings and inverse mappings in introductory differential geometry. The transition functions baffle me. Suppose our manifold of ...
1
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3answers
89 views

How to find the original function from a definite integral.

I have that $\int_{0}^{x} f(x) \,dx = 2x,$ and I would like to find $f(x)$. I am not even sure how to begin. I would appreciate any help!
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2answers
38 views

Possible values of z?

Let $f:[-2,2]\to \mathbb {R} $ where $$f(x)=x^3+2(\sin x)^5+3(\tan x)^7+\left\lfloor\frac{x^2+1}{z}\right\rfloor $$ is an odd function then what are possible values of $z$? $\lfloor\cdot\rfloor $ is ...
0
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1answer
66 views

Finding range of an algebraic fraction

How to find range of the function $f(x)=\frac{2x^4-14x^2-8x+49}{x^4-7x^2-4x+23}$ ? The question seems simple but I can't figure it out!
0
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1answer
35 views

How to correct values diverging from reality

I have two systems measuring the same thing: the state of charge (SoC) of a battery. The range is from 0 to 100%. One system seems to be correct, and shows a higher SoC of 30%, when the other systems ...
0
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1answer
36 views

Turn a function into an odd function?

I was asked to let: $$ f(x) = \begin{cases} 1-x, \text{if $0 \le x < 1$} \\ 0, \text{if $ 1 \le x < 2$} \end{cases}$$ Let fodd be the $4$ periodic odd extension of $f(x)$. Find the ...
1
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1answer
34 views

general solution function using Method of Characteristics

Suppose I am given a function $f(x, y, z)$ that is such that $3 f_x + xf_y + 2yf_z = 0$ I want to know how to write down a general representation for functions with such a property. Proceeding ...
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1answer
46 views

I have to find this Taylor series for function …

I'm helping a litle bit to my neighbour with "Calculus 1" and he sent me this exercise, which I sometimes remembered what trick you have to get, but now I forget a little bit. So, we have function $...
0
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1answer
52 views

How many different functions are there that are equal to their own inverse? [duplicate]

I know that functions can be their own inverse such as $f(x)=x$ however I thought there were only two $f(x)=x$ and $f(x)=-x$. Is there more?
-3
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2answers
337 views

Differentiation of a modulus function

How to find derivative of $$f(x)=|\sin^{-1}(2x^2-1)|$$ Please provide stepwise mechanism. The original question was to find domain of derivative of y=|arc sin(2x^2−1)|. My METHOD- My attempt was ...
1
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1answer
25 views

The significance of the 3-dimensional plot from homogeneous coordinates of a 2-dimensional function

If $(x,y)$ are your standard Cartesian coordinates and $(X,Y,Z)$ are homogeneous coordinates, then $x=X/Z$ and $y=Y/Z$. So if we have a function $f(x,y)$ we can convert it to a function $F(X,Y,Z)$ by ...
8
votes
2answers
109 views

Is there a simple sufficient condition for a function to depend “only on $r$”?

Suppose I were to pose the following problem to a class of calculus students: What is the magnitude of $\nabla f(x,y)$, where $f : \mathbb{R}^2 \to \mathbb{R}$ is the paraboloid function $$ f(x,...
1
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2answers
25 views

How find $\sum_{0 \leq k: \leq 100 \ gcd \left( k, 100 \right) =1} f \left( \frac{k}{100} \right)$ if $f \left( x \right) = \frac{9^x}{3+9^x}$ ?

How find of $\sum_{0 \leq k: \leq 100 \ gcd \left( k, 100 \right) =1} f \left( \frac{k}{100} \right)$ if $f \left( x \right) = \frac{9^x}{3+9^x}$ ?
0
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1answer
74 views

Sum of primes at minimal $\gt t!$

$$2+3+5+17+97+599\cdots a_t \gt t!$$ What does that mean? Well it is a sum that follows specific rules. For one, the number of terms in the sequence is $t$. Similarly, $a_t$ represents the $t$'th ...
9
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1answer
132 views

Non-computable function having computable values on a dense set of computable arguments

A rational complex number is a complex number whose both real and imaginary parts are rational numbers. Note that a rational complex number is a finitary object that can be an input or an output of an ...
3
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2answers
110 views

Closed-Form Modular Arithmetic

Is there a way to define modulo division (or functions of modular arithmetic in general) as superposition of (elementary?) functions? For example, the multiplication is first introduced as summation,...
1
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1answer
27 views

Is it true that $f(v) \circ T = f(v \circ T)?$

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function, and take $v \in L^1(\Omega)$. Suppose $T:\Gamma\to \Omega$ is a diffeomorphism between two bounded domains. Is it true that $$f(v) \circ T =...
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2answers
36 views

When is the inverse of a reciprocated function equal to the function?

If $f(x)=f(-\frac{1}{x})$, are there finite or infinite solutions for this? Can we tell? Thank you very much.
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6answers
2k views

What does it mean when two functions are “orthogonal”, why is it important?

I have often come across the concept of orthogonality and orthogonal functions e.g in fourier series the basis functions are cos and sine, and they are orthogonal. For vectors being orthogonal means ...
2
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1answer
42 views

Why is the digamma function close to 0 for large arguments?

I have taken the limit of both sides of an equation for x going toward infinity. There is a digamma (psi(x)) function on the RHS, and the limit of the term is supposed to be (at least close to) 0. ...
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0answers
45 views

Find formula that produces desired graph

Let's say we have an urn of balls of size $n$, Each ball has probability $p$ of being red. I take a sample from this urn without replacement and calculate the probability of having at least a ...
0
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1answer
43 views

On smoothness of a function

Consider the following function $f(x) = \ln \left( {\cosh \left( {{{\left[ {\max \left( {0,x} \right)} \right]}^n}} \right)} \right)$ where $max(.)$ is max function, $ln$ is natural logarithmic ...
5
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2answers
347 views

What is the geometric average of 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10…?

It is known that the Khinchin constant is not the geometric mean of the first $n$ coeffecients, as $n$ approaches infinity, of the continued fraction of e, which is $$[2; 1, 2, 1, 1, 4, 1,1, 6, 1, 1, ...
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1answer
81 views

Values of the Sudan function

I am talking about the first discovered recursive function which is not primitive recursive. I would like to know the exact values of $\ f(3,3,3), f(2,0,4), f(2,7,1), f(2,3,2)$ (where $f$ is the ...
1
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2answers
155 views

Finding Period From Functional Equation

How to find the period of real valued function satisfying $f(x)+f(x+4)=f(x+2)+f(x+6)$ ? Note:Use of recurrence relations not allowed.Use of elementary algebraic manipulations is better!
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1answer
172 views

Proving Odd & Even Functions

For the following: If $f(x)$ is an odd function, then $|f(x)|$ is _______. If $f(x)$ is an odd function and $g(x)$ is an even function then, $(f~o~g)(x)$ is _______. If $f(x)$ is an odd ...
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0answers
54 views

Can anyone explain these conclusions? Permutations, Symetric group…

The conclusions start off like this:I will highlight what is unclear in yellow. $sgn G$-sign of G permutation, $Ker$-kernel of a function Lets define the function: $\ \Phi$ like: $(\forall G \in S_n)...
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3answers
51 views

Comparison of three functions

How to prove that $$x-x^2<\ln(1+x)<x-\frac{x^2}{2(1+x)}$$ for all $x>0$?
6
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2answers
105 views

Attempt to prove: $f:[0,1]\to\mathbb R$, for every $c \in [0,1]$ the limit $ \lim_{x\to c} f(x)$ exists, so $f$ is bounded.

Let $f:[0,1]\to\mathbb R$ such that for every $c \in [0,1]$ the limit $ \lim_{x\to c} f(x)$ exists (and finite). Prove that $f$ is bounded. My attempt: For every $c \in [0,1]$ there exists $\delta_c ...
3
votes
3answers
650 views

The difference between 'solution' and 'root'

I am wondering about the difference between the following demands: Prove that P(x) has at least one root. Prove that P(x) has at least one solution. Are they the same? The background to ...
1
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2answers
33 views

Maxima of a function

Question: If $$f(x) = \cos\frac{\pi x}{2015}; x>0$$ and $$f(x) = 2x + a; x\leq 0$$ Find the values of $a$ such that $x = 0$ is a point of local maxima for $f(x)$ Attempt: As it's a ...
2
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1answer
113 views

Why is the definition of the absolute value $|x+1|$ the way it is?

In my notebook it is given that for the above function, we would have: $f(x) = {-(x+1), x<-1; (x+1), x\geq-1}$ What I don't get is why did we take $-1$ instead of $0$ as is the case for the ...
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votes
1answer
44 views

Coninutity of this function in interval $(0,1)$ [closed]

Let $f(x)$ be the function defined on the interval $(0,1)$ by $$ f(x) = \begin{cases} x(1-x) \quad\text{if}\quad x \in \Bbb Q \\ \frac{1}{4}-x(1-x) \quad\text{if}\quad x \in \Bbb ...
0
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1answer
47 views

Definition of a function error in textbook?

The excerpt from my books reads: "In a function, it's okay for two or more values of the independent variable to map to a single value of the independent variable. But it is not okay for a single ...
5
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2answers
361 views

Impossible numbers drawn from tricky function

The function is this: $f(\frac{a}{b},\frac{c}{d})=\frac{a+c}{b+d}$ where $0\lt \frac{a}{b} \lt 1$ $0\lt \frac{c}{d}\lt 1$ $a,b,c,d$ are all integers $a/b$ and $c/d$ are in lowest terms Are there ...
2
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2answers
806 views

Explaining why the absolute value of an odd function is even.

For the following: If $f(x)$ is an odd function, then $|f(x)|$ is _____. I said even, because I graphed an odd function and then the absolute value of it and ended up with an even function. The ...
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votes
4answers
131 views

Find the the value of [closed]

Given that $f(x)+3x f (\frac{1}{x})=2(x+1)$ Find the value of $f(101)$?
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3answers
105 views

Find the function $\hat{g}$ that maximizes $\int_0^1 x^2g(x)dx$ over the set of all functions that satisfy the following conditions:

Find the function $\hat{g}$ that maximizes $\int_0^1 x^2g(x)dx$ over the set of all functions that satisfy the following conditions: $\int_0^1 |g(x)|^2dx =1$, $\int_0^1 g(x)dx=0$, and $\int_0^1xg(x)...
0
votes
1answer
18 views

Equivalent definitions of an orthonormal function

I want to prove that the following two definitions for an orthonormal function $\phi$, in terms of $kT$ time shifts, are equivalent. So let $T$ the symbol period and $k$ an integer. Definition 1 ...
2
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1answer
36 views

The non-functional number

Lets say $f(1)=1, f(2)=2, f(3)=3,$ and $f(4)=n$. Some rules to follow are that: $1)$ $f(x)$ could literally be any function and it depends on what $n$ is. 2) $n$ is an integer. 3) If there is an ...
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0answers
43 views

Pairing function output that can be summed

Is there a pairing function that can take in a set of natural numbers N with a known set length and output a single natural number ...
1
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2answers
64 views

Multivariable Functions - Second Derivative Problem

So here is the problem: Calculate the second class derivative on $(1,1)$ of the equation $x^4+y^4=2$ I found this problem on my proffesor's notes. However it doesn't state whether a partial or a ...
0
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0answers
8 views

How to show monotonicity of this weighted average?

I have a sequence of positive and increasing (in all of its arguments) functions , $\{f_i(x_1, x_2, \dots, x_M)\}_{i=1}^M$. I also have: $$ \Omega_i=\frac{(x_i+f_i(\cdot))^{-1}}{\sum_i(x_i+f_i(\cdot)...
43
votes
5answers
3k views

Functions that are their own inversion. [duplicate]

What are the functions that are their own inverse? (thus functions where $ f(f(x)) = x $ for a large domain) I always thought there were only 4: $f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ ...
-1
votes
2answers
99 views

The range of the function $f(x,y)=(x+y,xy)$

I have the following homework question: $$\begin{split} f: \mathbb I \times \mathbb I &\to \mathbb R\times \mathbb R\\ f(x, y) &=(x+y, xy)\end{split}$$ Does there exist $(x, y) \in \mathbb ...