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

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2answers
521 views

Average Rate of Change f(x) involving sin

Can anyone check and tell me if 34/19 is correct - that's what I got it. If not can someone explain how to solve this. Thanks! :)
1
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3answers
263 views

Maximal domain for composite functions.

Question $\mathbf 5$ If $f:(-\infty,1)\to R$, $f(x)=2\log_{\,e}(1-x)$ and $g:[-1,\infty)\to R$, $g(x)=3\sqrt{x+1}$, then the maximal domain of the function $f+g$ is ...
0
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1answer
44 views

Differentiating by Quotient Rule

I don't understand why its raised to -1/2 (don't you have to subtract 1?) Can you differentiate by chain-rule? Yes/no and why?
0
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1answer
184 views

Maximal value of domain for a function by looking at inverse function.

The function g:[–a,a]→ R, g(x)=sin(2(x-π/6))has an inverse function.The maximum possible value of a is: From what I understand the domain of g(x) is the range of g'(x). So I would try to find the ...
0
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3answers
113 views

Derivative of $y=\ln(\tan^{-1}(2x^4))$

Find derivative of $y=\ln(\tan^{-1}(2x^4))$ The answer is: $y^{\prime}=\dfrac{8x^3}{(4x^8+1)\tan^{-1}(2x^4)}$ Can you show the steps on how to get this answer? I know that: ...
3
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1answer
60 views

Rigorous definition of relation composition

Let $R$ be an $n$-ary multivalued function on $A$, and let $S_1, ..., S_n$ be a list of length $n$, each member of which is an $m$-ary multivalued function on $A$. How does one rigorously define the ...
3
votes
1answer
161 views

Real analytic $f$ satisfying $f(2x)=f(x-1/4)+f(x+1/4)$ on $(-1/2,1/2)$

What are the real analytic functions $f\colon (-\frac{1}{2},\frac{1}{2}) \to \mathbb{C}$ that satisfy the following functional equation, and how are they derived? ...
3
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2answers
121 views

Derivative result is $0 \over 0$. Does it imply the point isn't differentiable?

$$f'(x) = {2x \over {4(x^2)^{3 \over 4}}}$$ for $x_0$ the derivative is $0 \over 0$, Which is defined as infinity as much as I know. Is it sufficient to say the above in order to prove $x_0=0$ isn't ...
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1answer
32 views

Rewrite the function as a cross product.

Good afternoon; How can I write this subset function as cross product. f : A → 2^B as a cross product. Regards,
5
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3answers
339 views

How to find the range of the function: $f(x) = \sqrt{x-1}+2\sqrt{3-x}$

Problem : Find the range of the function: $f(x) = \sqrt{x-1}+2\sqrt{3-x}$ Solution : Domain of this function can be determined as : $x - 1 >0 ; 3-x >0 \Rightarrow x >0 ; x <3 ;$ ...
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2answers
138 views

What do you call a function with the property $f(-x)=-f(x)$?

What is this property called? The domain and codomain of the function can be for example $\mathbb Z^n$, $\mathbb Q^n$ or $\mathbb R^n$ ($n>0$), potentially excluding the $0$ point. Examples: ...
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4answers
169 views

Check that two function $f(x,y)$ and $g(x,y)$ are identical

Given that $f(x)$ and $g(x)$ are two polynomials of degree $n$, we know that if we can find $n+1$ distinct numbers $x_i$, $i=1,\cdots,n+1$ such that $f(x_i)=g(x_i)$ then $f(x)$ and $g(x)$ are ...
0
votes
1answer
67 views

Is there a name for this kind of function?

I've learned a basic definition for function in calculus and I noticed that in: $$f(x)=[formula]$$ We take one number in the $LHS$ (which doesn't involve calculations, we just pick a number) and ...
0
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1answer
101 views

Representation of heaviside step functions

Can the heaviside step function, $u(t)$ be represented like so: $$u(t)=\frac{1}{2}\left(\frac{|x|}{x}+1\right)$$
0
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1answer
41 views

$M_1 = (x,y)\quad x²+y²+6y = 7 $ to $x \rightarrow y$

I have two relations: $$M_1 = (x,y)\qquad x²+y²+6y = 7 $$ $$M_2 = (x,y)\qquad x²+y²-6x = 7, \qquad y \ge 0$$ The question is if this relations also reflex functions like $x \rightarrow y$? I ...
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1answer
21 views

Function fromal definition

The relation $R:= \{(x,y) \mid y= \vert x\vert \} \subseteq \mathbb{Z} \times \mathbb{N}$ is a function, but the relation $R:= \{(y,x) \mid y= \vert x\vert \} \subseteq \mathbb{N} \times \mathbb{Z}$ ...
1
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5answers
1k views

What's the domain of $f(x) = \sqrt{x^2 - 4x - 5}$?

What's the domain of the function $f(x) = \sqrt{x^2 - 4x - 5}$ ? Thanks in advance.
4
votes
1answer
142 views

derivative calculation involving floor function $\left\lfloor {{x^2}} \right\rfloor {\sin ^2}(\pi x)$ [duplicate]

I was asked to find when the function is differentiable and what is the derivative of: $$\left\lfloor {{x^2}} \right\rfloor {\sin ^2}(\pi x)$$ Now, I am not sure how to treat the floor function. ...
1
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1answer
71 views

Continuous $f$ such that the set of translates of multiples of $f$ is a vector space of dimension two

How can we derive all of the continuous functions $f \colon \mathbb{R} \to \mathbb{R}$ such that $V=\{af_b : a,b \in \mathbb{R}\}$ is a vector space of dimension two, where $f_b\colon \mathbb{R} \to ...
1
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1answer
97 views

Find values of $\alpha$ for $ f:\left[0,1\right]\rightarrow\left[0,1\right] $ such that $ f\left(c\right)=\alpha\cdot c$

I'm having a bit of trouble with a homework question. Here it is: Let there be a function that $ f:\left[0,1\right]\rightarrow\left[0,1]\right] $ a continuous function. For what values of $\alpha ...
0
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5answers
69 views

What function, f(x), converges to linear function when x -> inf as well as when x -> -inf

I know that $\sqrt{1+x^2}$ does, but it is not good enough, because I need it to converge to $a+b x$ when $x\to\infty$ and $c+d x$ when $x\to-\infty$, with $a,b,c,d\in\mathbb{R}$. How to find the ...
2
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2answers
146 views

Simple Quadratic function exercise

I am a math teacher and I want to have some other opinions regarding an exercise made from one of my colleagues because I think that she was wrong when correcting the solutions from her students. The ...
1
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3answers
66 views

finding a limit of a general function. [duplicate]

${\rm f}$ is differentiable at ${\rm f}\left(1\right),\,$ ${\rm f}\left(1\right) > 0$. Calculate the following limit, and here's what I did: $$ \lim_{n \to \infty }\left[% {\rm f}\left(1 + ...
5
votes
2answers
120 views

Differentiable $f$ such that the set of translates of multiples of $f$ is a vector space of dimension two

How can we derive all of the differentiable functions $f \colon \mathbb{R} \to \mathbb{R}$ such that $V=\{af_b : a,b \in \mathbb{R}\}$ is a vector space of dimension two, where $f_b\colon \mathbb{R} ...
0
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1answer
17 views

Questions on Sierpinski's theorem, why $f_1(I\cap m(r))\subset f(I)$?

I'm reading the following post and I have a questions that I cannot resolve: In the last, or before last paragraph, of the above post the writer makes an observation: $$r\in f_1(I\cap m(r))\subset ...
3
votes
2answers
56 views

Set of non-decreasing function in bijection with R

I've learnt and understood the demonstration for "the set of all non-decreasing function is uncountable" with the diagonalization proof, but how could i demonstrate it is in bijection with R (the set ...
0
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3answers
74 views

Study of a function

I have to find the relative maximum and minimum points of this function $y=|x^2-4x|$ distinguishing stationary points from angular points. I tried to see this function as $y=x^2-4x$ as $x>0$ and ...
2
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5answers
61 views

More precise way of solving inequality

I need to solve this function: $$ \lvert x^2-1\rvert\ge 2x-2\\ $$ I solved this equation: For $x<0$, the solution is non existing, here I got negative root, when I tried to solve quadratic ...
0
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3answers
71 views

$f(x)$ is uniformly continuous $\Rightarrow f'(x)$ is uniformly continuous

$f(x)$ is uniformly continuous $\Rightarrow f'(x)$ is uniformly continuous I've encountered with an example which is basing on this statement. Is this statement true?
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2answers
102 views

Bijection from ordered pairs of $[0,n]$

I am looking for a simple expression to convert ordered pairs from $[0,n]$ to the first smallest subset of $\mathbb N$. For example if $n = 3$: $$ (0, 1) \rightarrow 0$$ $$ (0, 2) \rightarrow 1$$ $$ ...
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0answers
42 views

What is passivity property for a function?

I'm looking for info about the passivity property of a function. I've heard about this in a course of "Theory of systems and control"and given definition is: f continuos and derivable and with the ...
6
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1answer
83 views

Obtaining a binary operation on $X \rightarrow Y$ from a binary operation on $Y$. What, if anything, to make of this observation?

Let $X$ and $Y$ denote sets. Then if $+$ is a binary operation on $Y$, then we can obtain a new binary operation $+'$ on $Y^X$ in a canonical way as follows. $$(f+' g)(x) = f(x)+g(x)$$ Question. The ...
1
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1answer
25 views

Question on Sierpinski's theorem, why $f_1,f_2$ are well defined?

I'm reading the following post and I have a question that I cannot resolve: Why are the function $f_1$ and $f_2$ are well defined? In other words: why $m(r)\cap n(r)=\emptyset$?
8
votes
2answers
189 views

Continuous $f$ satisfying $f(2x)=f(x-1/4)+f(x+1/4)$ on $(-1/2,1/2)$

What are the continuous functions $f\colon (-\frac{1}{2},\frac{1}{2}) \to \mathbb{C}$ that satisfy the following functional equation, and how are they derived? ...
0
votes
1answer
92 views

Is this conditional expectation an increasing function?

let $g(x)=E[Y\mid Y<x],$ where $Y=\max(Z_1,\ldots, Z_n)$ and each $Z_i$ is i.i.d. with density function $f(z)>0$ for any $z$ in some interval $(0,a)$, $a>0.$ Is $g(x)$ increasing in $x$, for ...
0
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1answer
48 views

Scoring algorithm for a Game

I have three variables: Good points = You get 1 good point every time you get a question correct. The maximum good points you can obtain is 21. Bad points = You get 1 bad point every time you ...
0
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1answer
35 views

Write Dickman Function as CDF

Is it possible to write the Dickman Function $F(\alpha)$ & $G(\beta)$ in terms of a cumulative distribution function? I would like to play around with the variance, etc as I am interested in how ...
3
votes
1answer
501 views

Bijection between [a,b) and [a,b] intervals. [duplicate]

I need help with finding a bijection between these two intervals: [a,b) and [a,b]. It is told that a,b are in R (real numbers). I know how to construct a bijection if a,b are integers, but i have no ...
0
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1answer
54 views

Adding together curves or shapes to approximate something more complex

I'm looking for proper terminology / references for the following sort of problem: Say we have some one-dimensional curve like $y = 10$ defined over the real valued domain $[0,1]$, and we ask, how ...
2
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2answers
63 views

How to find inverse of function $f(x, y)$?

I am aware of the method to find inverse function $f^{-1}(x)$ of $f(x)$, which is Replace $f(x)$ with $y$ Switch $x$'s and $y$'s Solve for $y$ Replace $y$ with $f^{-1}(x)$ the above method ...
5
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2answers
124 views

Solving $f(x+y)-f(x)=yf'\Big(x+ \dfrac y{2}\Big),\forall x,y\in \mathbb R$

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f(x+y)-f(x)=yf'\Big(x+ \dfrac y{2}\Big),\forall x,y\in \mathbb R$ , then how do we show that $f$ is a polynomial of degree at ...
1
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0answers
176 views

Associative, commutative properties and identity elements of non-binary functions

I'm making a compiler (for a new language) wich supports AC unification via pattern matching. The matching algorithms already works but i'm having trouble with the logical and mathematical aspects of ...
0
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2answers
76 views

If $f(x) = \sin \log_e (\frac{\sqrt{4-x^2}}{1-x})$ then find the range of this function.

Problem : If $f(x) = \sin \log_e (\frac{\sqrt{4-x^2}}{1-x})$ then find the range of this function. My approach : $\frac{\sqrt{4-x^2}}{1-x} >0 \Rightarrow 1-x >0 $ also $4-x^2 >0$ ...
2
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5answers
404 views

Help with finding tangent to curve at a point

Find an equation for the tangent to the curve at $P\left( \dfrac{\pi}{2},3 \right )$ and the horizontal tangent to the curve at $Q.$ $$y=5+\cot x-2\csc x$$ $y\prime=-\csc ^2 x -2(-\csc x \cot ...
3
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3answers
295 views

Is $\log (1 + {x^2})$ uniformly continuous on $[0,\infty)$? [duplicate]

Is $\log (1 + {x^2})$ uniformly continuous? Here is my attempt: Let $\forall\left| {x - y} \right| < \delta$: $\left| {\log (1 + {x^2}) - \log (1 + {y^2})} \right| = \left| {\log (\frac{{1 + ...
2
votes
5answers
247 views

If Bob and Alice never met in class, at least one of them missed at least half of the classes

Alice opened her grade report and exclaimed, "I can't believe Professor Jones flunked me in Probability." "You were in that course?" said Bob. "That's funny, i was in it too, and i don't ...
0
votes
0answers
31 views

Show properties of some general function

I've got some function $f : P ( \mathbb{N} ) \rightarrow P(P ( \mathbb{N} ) \times P ( \mathbb{N} ))$ Described as $f(Z) = \{\langle X,Y\rangle: P ( \mathbb{N} )\times P ( \mathbb{N} ) \ \mid Z ...
1
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1answer
225 views

supremum and infimum of a bounded and decreasing sequence

Is there supremum and infimum of a bounded sequence? I have a bounded and decreasing sequence. Why does this sequence have infimum?
1
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2answers
98 views

Prove/Disprove $f(x) = x + \frac{x}{{x + 1}}$ is uniformly continuous at $\forall x,y \in [0,\infty )$ [duplicate]

Prove/Disprove $f(x) = x + \frac{x}{{x + 1}}$ is uniformly continuous at $\forall x,y \in [0,\infty )$ This is my trial: $$\forall \varepsilon > 0\exists \delta > 0.\forall x,y \in ...
1
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2answers
210 views

Limits involving trigonometric functions $f(x)=\lfloor{x^2 \rfloor} \sin^2(\pi x)$

I was asked to find for what values of x the function is differentiable and write down the derivative. $f(x)=\lfloor{x^2 \rfloor} \sin^2(\pi x)$ for $x \geq 0$. There are two steps: When $x \in ...