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

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0
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0answers
7 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: $$ ...
41
votes
5answers
3k views

Functions that are their own inversion.

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
84 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 ...
6
votes
3answers
267 views

What is the name for a function whose codomain and domain are equal?

What do we call a function whose domain and co-domain are the same set? Edit: While i expressed my question in terms of functions, domains and codomains, i was actually interested in the most ...
2
votes
1answer
52 views

What does it mean to convolve matrices of finite dimension?

If one is given two matrices $I$ and $K$ what does the notation: $$ I * K $$ mean rigorously/precisely? I do know the definition of convolution: $$ s[i, j] = (I * K)[i, j] = \sum_m \sum_n I[m,n] ...
0
votes
1answer
73 views

Evaluating function

Here is the function: $$f(a)=\sqrt{f(a)+\sqrt{f(f(a))+\sqrt{f(f(f(a)))+\cdots}}}$$ Is there another way to represent this function so that it only has $f(a)$ on one side and no $f(a)$'s on the other ...
1
vote
1answer
52 views

Mapping the intersection of hyperplanes/simplex to lower-dimensional unit-simplex

Suppose I have an object in $\mathbb{R}^5$ described by: $$x_1+x_2+x_3+x_4+x_5=1$$ $$x_1+2x_2+3x_3+4x_4+5x_5=6$$ $$x_1+7x_2+8x_3+9x_4+10x_5=11$$ $$x_1,x_2,x_3,x_4,x_5 \geq 0$$ Is there a way that I ...
0
votes
0answers
33 views

Is there a name for linear transformations $T: C(\mathbb{R}) \to \mathbb{R}^n$

Briefly, I wish to study forms that can approximate all such linear transformations $T: C(\mathbb{R}) \to \mathbb{R}^n$. Clearly the Riesz representation theorem cannot be applied in this case as ...
6
votes
2answers
222 views

Showing a function is invertible

I came across this problem and not sure how to prove it. Show that if $ f\circ f \circ g\circ g \circ f\circ f $ is invertible then $ g $ is invertible. I'm not sure if it's correct to say that ...
0
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1answer
22 views

I am trying to verify a sufficiency condition for one concave function being greater than another

Let $f,g:[a;b]\rightarrow R$ be two functions that are strictly concave and strictly increasing on $(a;b)$. Moreover, $f(a)>g(a)$ and $f(b)>g(b)$. Finally, let $x\in(a;b)$ such that ...
0
votes
2answers
52 views

Prove there exists a bijective function $\left\{a,\cdots,b\right\}\rightarrow\left\{a+k,\cdots,b+k\right\}$ for $k\in\mathbb{N}$

I must prove that there exists a bijective function $\left\{a,\cdots,b\right\}\rightarrow\left\{a+k,\cdots,b+k\right\}$ for $k\in\mathbb{N}$ (this is not homework). This is the proof I've come up with ...
0
votes
0answers
30 views

Find all real functions $F(x)$ so that $F(x_1) - F(x_2) \le (x_1 - x_2)^2$ for any choice of $x_1,x_2 \in \mathbb{R}$

Find all functions $F(x) : \mathbb{R} \rightarrow \mathbb{R}$ such that for any $x_1, x_2$ the following holds: $F(x_1) - F(x_2) \le (x_1 - x_2)^2$ Sketch a proof of why these are the only such ...
-2
votes
2answers
55 views

Function undefined at non-integer values [closed]

Is there a function $f(x)$ which is not defined at integer values? Please do NOT answer $f(x) = \begin{cases} a, & \text{if } x \in \mathbb{Z}, \\ \text{undefined}, & \text{otherwise} ...
0
votes
1answer
16 views

why is $\overline{\text{span}\{E_x(C_0(X))\mathbb{C}\}}=\mathbb{C}$

Let $X$ be a locally compact Hausdorff space, $C_0(X)$ the vector space of complex valued functions vanishing at infinity, endowed with the supremumsnorm. For $x\in X$ consider the ...
0
votes
2answers
26 views

If $f$ is continuous and piecewise $C^1$ and $f'$ is bounded a.e., is $f$ Lipschitz?

If $f$ is continuous and piecewise $C^1$ on $\mathbb{R}$ (only a finite number of pieces) and $f'$ is bounded a.e., is $f$ globally Lipschitz? So $f$ is only not differentiable in a finite number of ...
1
vote
1answer
32 views

Geometric Interpretation of a function

Look at the following functions: $$l(x)=x/\sqrt{1+x^2}$$ $$k(x)=x/\sqrt{1-x^2}$$ These functions give a homeomorphism between $\mathbb{R}$ and $(-1,1)$. Can someone give a geometric interpretation of ...
6
votes
2answers
157 views

Interesting functional equation: $f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$

Solve for the function f(x): $$f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$$ I'm not able to solve this. [For instance, I tried solving for $f(\frac{x}{x+f(x)})$, but this doesn't lead me ...
1
vote
1answer
24 views

Order of solving a function.

What is the proper order to solve functions and their inverses? Is it different from working across the equal sign in an normal algebra problem, as in is there a special way just because its a ...
1
vote
1answer
31 views

Domain of the definition of a composite function

If $$f(x)= \sqrt{3|x|-x-2} \\ g(x)=\sin(x),$$ then the domain of definition of $(f\circ g)(x)$ is ...? How do you calculate the domain of a composite function like this?
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1answer
41 views

$f : A \to B$ s.t. for all $x, y \in A, x R y \iff f(x) S f(y)$

Theorem. A relation $R$ on a set $A$ is reflexive and transitive if and only if there is a set $B$ with a partial order $S$ and a function $f : A \to B$ such that for all $x, y \in A, x R y \iff ...
4
votes
2answers
100 views

Progression of the reciprocal of squares $ \lt \frac{1}{4}$

$$\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}+\frac{1}{121}\cdots \lt\frac{1}{4}$$ This is an interesting summation in which the addition of the next term must make the sum $\lt\frac{1}{4}$. ...
1
vote
3answers
55 views

Prove this function is injective $f(x)=x+\mod(x,7)$

Prove this function is injective $f(x)=x+\mod(x,7)$. Attempt: I tried separating in two cases: $x \equiv y \pmod 7$ and $x \not \equiv y \pmod 7 $: First case: $$f(x)=f(y) \iff x+\mod(x,7)=y+ \mod ...
-5
votes
1answer
47 views

Properties of Functions and Intervals [closed]

Recall that any interval of the form $(a, b)$, $(a,\infty)$, or $(−\infty, b)$ (where $a$ and $b$ are real numbers and $a < b$) is called an open interval. Any interval of the form $[a, b]$, ...
0
votes
1answer
31 views

Standard notation for brackets around trigonometric arguments

I've been going through this site quite a lot lately and I've noticed one thing, the notation $\sin(x), \, \cos(x), \, \tan(x),\ldots \,$ is used frequently. Now, I do tend to bracket the arguments ...
2
votes
2answers
99 views

Can I soundly define a function which maps to itself?

A function can be defined by specifying a set of tuples. If I write the definition of a function $f = \lbrace(0, f) \rbrace$, is this function sound? Will this lead to a paradox? The domain of this ...
2
votes
1answer
37 views

Parabolic range conditions proof

This problem is getting the better of me, since I have no idea where to start: The equation of a curve is $y=ax^2-2bx+c$, where a, b and c are constants with $a>0$. Given that the vertex of the ...
3
votes
1answer
48 views

Set Notation with exponent

I am looking at the function: $$f: \{5\}^2 \to \{5\}$$ it is certainly nothing too exceptional , but I find it difficult to understand what $\{5\}^2$ as a set notation and from then the whole ...
0
votes
2answers
49 views

Is this function monotonically decreasing?

Fix $t \leq T$. $$f(n):= \frac{e^{nt}-e^{-nt}}{e^{nT}+e^{-nT}-2}$$ Is $f$ monotonically decreasing in $n$? I tried derivating it but it didn't really help me.. please help me out.
-1
votes
2answers
107 views

How to prove that $1+\frac{x}{1!}+ \cdots +\frac{x^{n}}{n!}=0$ cannot have repeated roots?

Using Rolle's Theorem: How to prove that $1+\frac{x}{1!}+ \cdots +\frac{x^{n}}{n!}=0$ cannot have repeated roots? Well intuition might help a bit,but I can't figure out a way using Rolle's theorem. ...
0
votes
1answer
34 views

Determining position with respect to time

If we assume $s(t)$ as a time-dependent position function and $v(t)$ as a time-dependent velocity function($v = \frac {\mathrm{d}s} {\mathrm{d}t}$) and $v=8\sqrt{s}$, how could I determine $s$ with ...
1
vote
5answers
122 views

Can the roots of $f(x)=x^4-x^3+2x^2-x-1$ be found algebraically?

Can the roots of $f(x)=x^4-x^3+2x^2-x-1$ be found algebraically? Are there multiple methods for doing so?
1
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0answers
35 views

Optimize multi-step calculation into one step?

Prerequisites The following helper functions are defined: Condition $$ \delta (x,y) = 0^{(x-y)^{2}} $$ This is a simple condition, it returns 1 when x = y. It will be used to modify products to ...
0
votes
3answers
61 views

Solving the functional equation $f(a + b) = a + f(b)$

How would you solve: $$f(a+b)=a+f(b) ?$$ It seems similar to the Cauchy equation $$f(a+b)=f(a)+f(b),$$ but I'm not sure what to do with this. I have a feeling the only solution is $f(k)=k$ but idk. ...
0
votes
0answers
38 views

Trying to prove that if $f:[a, b]\to[s, t]$ is monotone then $f$ is continuous

I'm trying to prove that if $f:[a, b]\to[s, t]$ is monotone (and its image is closed interval) then $f$ is continuous. My attempt: I say wlog, $f$ is increasing. I know that a monotone function only ...
0
votes
0answers
20 views

Local minima or maxima of conditional function

How do I find the local minimum/maximum of this function? $f:{R-2}\to{R}$ $f(x)=\frac{1}{x-2}e^{\left|x\right|}$ I wrote it like this $$f(x)=\left\{\begin{array}{cc} \frac{1}{x-2}e^x & x \ge 0 ...
1
vote
2answers
414 views

Can any function represent something in the real world?

We know that the volume of a cube can be represented by the function: $V(x)=x^3$, where $x$ is side length. $x^2$ can represent the volume of some material that has a constant side ($1$). The function ...
1
vote
1answer
57 views

Finding the minimum value of a function.

Find the minimum value of the function: $$f(x) = \frac{\left(x+\frac{1}{x}\right)^6-\left(x^3+\frac{1}{x^3}\right)^2 - 2}{\left(x + \frac{1}{x}\right)^3 +\left(x^3 + \frac{1}{x^3}\right)}$$ for ...
2
votes
1answer
35 views

Graph of a non-continuous function is closed

Exercise. Let $\ f\colon\mathbf R\to\mathbf R$ be defined by $$f(x)=\begin{cases}\frac{1}{x},\ x>0\\0,\ x\leq 0\end{cases}$$ Prove that the graph $\Gamma_f:=\{(a,f(a)):a\in\mathbf R\}$ of ...
0
votes
2answers
46 views

Unbiased estimator.

With sample variance defined as $S^2 = (\sum_{i=1}^n (X_i - \overline{X})^2 )/(n-1)$ A. Show that $E(X_i^2) = \sigma^2 + \mu^2$ using the fact that $\sigma^2 = E((X_i - \mu)^2)$
1
vote
6answers
66 views

Minimizing a function - sum of squares

I'm hoping you can help with this problem. I haven't taken calculus in years and I don't know where to start... The sum of squares of a sample of data is minimized when the sample mean is used as the ...
2
votes
1answer
83 views

Find all function $f: \mathbb{Q} \to \mathbb{Q}$ such that $f(x+f(x)+2y)=2x+2f(f(y))$ [closed]

Find all function $f: \mathbb{Q} \to \mathbb{Q}$ such that $f(x+f(x)+2y)=2x+2f(f(y))$ for all $x,y \in \mathbb{Q}$.
0
votes
1answer
46 views

Differentiability issue with this function

$f:D\to{R}$ $$f(x)=\frac{1}{x-2}e^{\left|x\right|}$$ Find the domain $D$ of the function and study whether the function is differentiable. Find the left and right derivatives in the points where the ...
0
votes
1answer
36 views

Can anyone help out with this proof, certain steps are unclear. Norm of linear operator.

I have the following norm defined as follows (in $R^n$, $x=(x^1,x^2,\ldots,x^n)\ $)$\| x\|_1= \sum_{i=1}^{n}|x^i|$ Let $A:R^m \to R^n$ a linear map of the spaces $(R^m ,\| \cdot \|_1 )$ and $((R^n ...
-5
votes
1answer
59 views

Why is this statement true? [closed]

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be the function $f(x,y)=(y-x^2)(y-2x^2).$ Why is this statement true: $t\mapsto f(t\xi)$ has in $t=0$ a local minimum for every $\xi\in\mathbb{R}^n$
0
votes
1answer
28 views

Each non-empty interval $I \subset \mathbb{R}$ we have $f(I)=f(\mathbb{R})$ when $f$ is additive

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an additive function such that $f(1)=0$. Then for each non-empty interval $I \subset \mathbb{R}$ we have $f(I)=f(\mathbb{R})$. Since $f$ is additive ...
0
votes
1answer
26 views

Determine relation of $x$ and $y$ from results

I can't seem to determine the relation between $x$ and $y$ for this problem. All of the previous ones I have done have been doable simply by eye-balling the relation between $x$ and $y$, but here I am ...
0
votes
2answers
40 views

What is the meaning of the notation of function

More specifically, what is meant by the function $T: \mathbf V \to \mathbf W$? I saw it in the discussion of linear map in Axler's Linear Algebra Done Right but could not understand this notation. ...
2
votes
4answers
120 views

What is the difference between these two statements involving minimums of a function?

All values at which $f$ has a local minimum. and All local minimum values of $f$.
3
votes
2answers
38 views

Expressing a function in terms of compositions of three functions.

Express the function F in the form $f \circ g \circ h$. $$F(x)=\frac {9}{( x^2 + 7)}$$ I'm not sure how to get $x^2+7$ in the denominator. Here is what I tried: $$h(x) = (x+7)$$ $$g(x) = x$$ ...
0
votes
3answers
40 views

Possible values of 'a' ? $f(x)=(x^2+ 2 ax +a^2-1)^{\frac{1}{4}}$

If $$f(x)=(x^2+ 2 ax +a^2-1)^{\frac{1}{4}}$$ has its domain and range such that their union is set of real numbers,then what should be the possible values of a? What can be the approach?