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

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1answer
32 views

Finding the domain of this trigonometric function

how can I find the domain of this function? f(x) = (xsin(x) + cos(x) / 1 - cos(x)) + (|X| - 2 / x^2 -4) I assume we don't want the dominator to be zero so f(x)1 ...
0
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1answer
52 views

Find the range of function $f(x) =\cos(\sin(\ln(\frac{x^2+e}{x^2+1})))+\sin(\cos(\ln(\frac{x^2+e}{x^2+1})))$…

Problem : Find the range of function $f(x) =\cos(\sin(\ln(\frac{x^2+e}{x^2+1})))+\sin(\cos(\ln(\frac{x^2+e}{x^2+1})))$ My approach : maximum value of the function is when denominator term is ...
0
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1answer
46 views

Prove concavity without testing the second derivative..

Consider a function $F(L)=(L-L^2a)T^{L-1}$, where $0<L<\frac{1}{a}$. The constants $a$ and $T$ may take values over $]0,1[$ and $[0.01,0.1]$, respectively. The first derivative of $F$: ...
2
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6answers
109 views

In f:A→B, surely A and B are redundant

If a function is a set of ordered pairs, it defines its own proper domain and codomain. f determines A; a minimal B. If we extend B, do we have a different function?
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1answer
45 views

Alternative Definition for Injective Function?

I came up with an alternative definition of an injective function and would like to know if it's correct and how to prove it if it is, or why it's not correct if it isn't. f:A→B is injective if ...
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0answers
39 views

Why is the function defined by $f(x_1,x_2)=0$ when $x_1=0$, and $x_2$ when $x_1\neq 0$ not regular?

I'm having trouble understanding what should be a straight forward example. Suppose $X\subseteq\mathbb{A}^2_k$ is cut out by the equation $x_1(x_2^2-x_1)=0$. Define a function $f:X\to k$ (here ...
1
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2answers
87 views

Why is $x^5 \sin x$ an odd function?

Why is $x^5 \sin x$ an odd function? Is the result just wrong? Because $f(-x)= (-x)(-x)(-x) \sin(-x) = (-x)(-x)(-x)(-x)(-x) (-\sin x) = (-x^5)(-\sin x) = x^5 \sin x$
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2answers
50 views

Showing one-one onto

I wanted to find the values of $a$ for which the function $f:\mathbb R\to \mathbb R$ defined by $f(x)=ax+\sin x$ is bijective. Any hint will be appreciated.
1
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1answer
80 views

inclusion of sets and inverse function

Is it true that for any function $f$, and sets $S_1,S_2$ such that $f:S1\rightarrow S2$, if g is the inverse of f $g = f^{-1}$, then $f(g[S1])\subseteq S1\subseteq g(f[S1])$?. If yes, is there a ...
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2answers
30 views

Range of a composite function

How to find the range of functions like $f(x)=\sin (x) ^{sin(x)}$ on $(0,\Pi)$? Usually, I find the inverse and then find the domain of the inverse function for the range of the original function, ...
1
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2answers
174 views

Transpose of composition of functions

I am trying to find an alternative proof that $(AB)^t=B^tA^t$. I think it is possible to do by showing that: $(g \circ f)^t=f^t \circ g^t$ where f,g are linear maps between vector spaces. I am ...
3
votes
7answers
150 views

$f\left(x + \frac1x\right)= x^3+x^{-3},$ find $f(x)$

$$f\left(x + \frac1x\right)= x^3+x^{-3},$$ find $f(x)$. What i do know at this state is that.. express x as a function of y : $y= x + 1/x$ $x^2−xy+1=0$ Quad formula: $x= (y ± \sqrt {y^2-4}) / 2$ ...
0
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4answers
57 views

Proving an equation is a fuction

Prove that the equation $y^3 + 3xy -5x^3 + 1 = 0$ defines $y$ as a function of $x$ for all $x$ in the real numbers.
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2answers
61 views

How to prove functions are odd and even

Show that any function f on [-a,a] where a is a positive constant, can be written as the sum of an even and an odd function?
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2answers
58 views

Notational difference, functions and mappings, talking about sets and classes

A Function is a set of pairs such that no two pairs have the same first member. My question summarized: What if I want to consider proper classes of pairs? The closest question to mine I could ...
7
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2answers
112 views

Function composition: $f^{653}(56)=?$

Let $f(x) = \frac1{(1-x)}$. Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$. Find $f^{653}(56)$. What I've done: I started with r=1,2,3 and noticed the following pattern: $$f^r(x)= ...
0
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1answer
70 views

Linearization of a function in the point 0, 0

The linearization of the function $ f(x, y) = 1 + 2(x + 1) + 3(y + 1) + 4x^2 + 5y^2 $ in the point (0, 0) is given by: $ L(x, y) = 6 + 2x + 3y $ I know this is true, but how does one come to this ...
1
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3answers
23 views

For which of the following functions the $\sum_{x\in S(f)}\frac{1}{x}$ converges?

For real valued function $f$ define $$S(f)=\{x:x>0,f(x)=x\}$$ For which of the following functions the $\sum_{x\in S(f)}\frac{1}{x}$ converges? $\tan x,\tan^2x,\tan{\sqrt{x}},\sqrt{\tan x},\tan ...
1
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1answer
31 views

Composite Functions

$f(x)= \dfrac{1}{10x+17}+13$ $g(x)= \dfrac{1}{9x-6}$ I need to find $f(g(x)).$ How do I do this? I keep on getting it wrong. The correct answer is $\dfrac{1998x-1202}{153x-92}$. But I am unsure how ...
0
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1answer
35 views

A question regarding real valued function

I have a question regarding real-valued function: Which of the following cannot possibly be the rule of any real-valued function? A) $y=\sqrt{x-1}$ B) $y=\sqrt{x-1}+\sqrt[3]{2+x}$ C) ...
1
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1answer
40 views

Finding inverses of two functions and their compositions to solve for unknown.

$$f(x) = 23x + 27,\;\; g(x) = 12x - d$$ I've found $f^{-1}(x),$ and $\,g^{-1}(x)$, but I don't know how to solve for $d$, given $$f^{-1}(g^{-1}(x)) = g^{-1}(f^{-1}(x)).$$ How do I do this please?
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2answers
31 views

Which of the following is constant?

If $f,g$ are continuous real valued functions such that $f\circ g$ is constant then which of the following must be constant? $$f,g,g\circ f$$ I think when $f\circ g$ is constant then at least one of ...
2
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1answer
60 views

Prove whether a particular function is concave

Given the following equation: $$V(w) = - \frac{\alpha}{2} \left[ y_1(w) + y_2(w) + \int _{-\infty}^{+\infty} \vert y_1(w) - y_2(w) - x\vert f_{T1}(x)dx\right] \\- \beta \int _{w - y_1(w)} ^{+\infty} ...
2
votes
3answers
294 views

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 ...
1
vote
1answer
55 views

Why have we made a function to be many to one and not one to many? [closed]

We have allowed function to only relate many to one but not one to many. Why haven't we included sin(x) to be a function? Is it just for simplicity? Also, I've seen someone quote a function not even ...
2
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0answers
27 views

Real Analysis Differential functions

I am currently working through an exercise set and I am a little stuck on the following question: For $a > 0$, define a function $f_a(x)= \begin{cases} x^a \sin(1/x), &x \ne 0\\0, &x=0 ...
3
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1answer
253 views

How to find the function $f$ that satisfies $f(x, y) = f(x^{-1}, y^{-1})^{-1}$ and $f(x, y)$ is $\approx$ $average(x, y)$?

Fist of all, I'm a programmer, not a mathematician, and I'm sorry for my non native English. And I'm sorry if the question is not appropriate, it is my first time here. Or if the question has no ...
1
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1answer
76 views

Lyapunov exponent for simple functions

Context: We know that $\cos(x)$ if taken recursively on itself, converges to the Dottie number, which is the function's stable fixed point then. On the other hand, for a function like $f(x)=3x$, ...
1
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1answer
34 views

Drawing toral automorphisms

How do we set about drawing a toral automorphism as in figure 5.1 in the picture above. How do we know where the points highlighted in yellow are? What happens if the eigenvalue (Im guessing some ...
0
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3answers
29 views

Limit of a function w

If $f(x, y)$ is a continus function, defined in whole $\mathbb R^2$, then the limit $$\lim_{(x, y)\rightarrow(2,2)}f(x, y)(x-1)(y-2) $$ The solution is $0$, but how? A very elaborative explanation ...
5
votes
2answers
212 views

Making a piecewise function continuous and differentiable at point

Problem: Let $f(x) = \left\{ \begin{array}{lr} \frac{\arctan(x)}{(1+x)^2} & : x \geq 0\\ Ae^x + B & : x < 0 \end{array} \right. $ Find $A$ and $B$ such that the function is ...
4
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6answers
77 views

Domain of $\frac{1}{\frac{1}{x}}$

Let $f(x)=\frac{1}{x}$, then we have $f^{-1}(x)=\frac{1}{x}$. So $f(f^{-1})=\frac{1}{\frac{1}{x}}=x$. My question is, what is the domain of $f(f^{-1})(x)$? is it everything? or everything but zero? ...
1
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3answers
357 views

Intersection of inverse images

Given $A$ and $B$ is the subset of $C$ and $f:C\mapsto D$, $$f(A\cap B)\subseteq f(A) \cap f(B)$$ and the equality holds if the function is injective. But why for the inverse, suppose that $E$ and ...
0
votes
1answer
154 views

Integral of a normal function multiplied by heaviside and delta functions

$\int_{-\infty}^{\infty} e^{2t}u(\tau - t)t^{2}\delta(t)dt$ Hi! How would I go about computing this integral? I understand I can change one of the integration limits and eliminate the heaviside ...
0
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0answers
31 views

Questions about functions, their domains and codomains.

I am playing around with equations about functions in general and have some questions. Question 1 If I have some functions $f,g\colon X^2 \rightarrow Y$ such that $f(x,y) = g(x,c)/g(y,c)$ then can ...
0
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1answer
32 views

Composing Piecewise Functions

I was wondering how to compose piecewise functions. On a practice exam I was reading, a question asks what F(F(x)) will look like if F(x)= 2x if x<1/2 and = 2-2x if x>=1/2. Would I just ...
1
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1answer
72 views

How to calculate “general” integral $\int\limits_{a}^{b}f(x)^2dx$?

How to calculate "general" integral: $\int\limits_{a}^{b}f(x)^2dx$?
0
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1answer
48 views

What is this integration “method” name?

I see that people often write this equality: $$\int\limits_a^bf(x)\,\mathrm dx=\int\limits_{f(a)}^{f(b)}f(x)\,\mathrm df(x)$$ when dealing with functins in general, that is when something is trying ...
1
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1answer
35 views

Inequality from a property of convexity

Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function. How can I prove that for each $x$, there is $c$ such that $f(x)+c(y-x)\leq f(y)$ for all $y$? One of the difficulties to solve is $f$ does not ...
0
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1answer
45 views

Empty function, what is it?

I meet with term 'empty function' from time to time. It's high time to understand its nature. What is field( set of arguments) and what is image? ( set of value)?
1
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1answer
51 views

Who is growing faster?

I am trying to prove that $\lim_{n\to \infty} { 2^{n^2} \over n!} = \infty$. I can't use l'Hôpital's rule (or I dnon't know how) and I don't recall any other method which could help me. It also isn't ...
0
votes
2answers
48 views

Limit as x approaches 0 from the left: $\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$

Help me find the limit as x approaches 0 from the left: $$\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$$ Thanks,
1
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1answer
18 views

Can the induced function of non-zero $f \in R[X]$ be zero, when $R$ is an infinite non-integral domain?

Let $R$ be an infinite commutative ring with $1$ which is not an integral domain. Is it possible to have a non-zero $f\in R[X]$ such that the induced map $\bar{f}: R \to R$ is zero? Please give ...
0
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2answers
265 views

Subspaces and annihilators

Suppose $V$ and $W$ are subspaces of a finite-dimensional vector space $U$. (a) Show that $W \subset V$ if and only if $V^0 \subset W^0$. (b) Show that $(V \cap W)^0 = V^0 + W^0$. (Hint: ...
2
votes
4answers
70 views

Find $\,\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$

How do I calculate $\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$. Please help me. Thanks!
4
votes
3answers
266 views

What is the opposite category of $Set$?

In $Set$ the initial object is the empty set, and it has an unique morphism to each other object, namely $f=\emptyset$. However I find it difficult to think about the category ${Set}^{op}$, is there ...
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1answer
27 views

Number of different functions

Suppose that $A$ has exactly $m$ elements and $B$ has exactly $n$ elements. How many different functions are there from $A$ to $B$? The answer is given by $n^m$ but i don't know how to get that And ...
1
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2answers
247 views

Unbounded function on compact interval?

So what are some unbounded function on compact interval, if there is any? Also, is the function $f:[0,\infty) \to \mathbb R$, $f(x)=x$ continuous?
2
votes
2answers
77 views

Does there exist a bijective mapping of an open interval with the corresponding closed interval having only finitely many points of discontinuity?

Given $a<b$, is there a bijection $f \colon [a,b] \rightarrow (a,b)$ such that $f$ be continuous except at finitely many points only? I know that there does exist a bijection of $[a,b]$ with ...
0
votes
2answers
39 views

Find $\lim_{x\to\infty} \frac{e^{2x}-1}{e^{2x}+1}$ and $\lim_{x\to-\infty} \frac{e^{2x}-1}{e^{2x}+1}$

How do I calculate $\displaystyle \lim_{x\to\infty} \frac{e^{2x}-1}{e^{2x}+1} \ , \ \lim_{x\to-\infty}\frac{e^{2x}-1}{e^{2x}+1}$. Please help me. Thanks!