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

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7 views

Is it possible to approximate $cos(x)$ with a linear combination of Gaussians $e^{-x^2}$?

I am interested in approximating $\cos x$ with a linear combination of $e^{-x^2}$. I am not an expert in approximation theory but there are a couple things that give me a bit of hope that it might be ...
0
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1answer
34 views

Confused by one-to-one question, I think it's order incorrectly

I have this question and it seems a tad redundant If $A$ and $B$ are infinite sets, is it possible for there to be a 1-1 function from $A$ to $B$ and a 1-1 function from $B$ to $A$ without there ...
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4answers
46 views

Why is this function a bijection?

Consider the function below $$f:\mathbb{R^+} \to \mathbb{R^+}$$ given by $$f(x) = \sqrt{x}$$. Now it makes sense that the function is injective because $f(x) = f(y) \implies \sqrt{x} = \sqrt{y} ...
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1answer
93 views

How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
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1answer
38 views

Lazy mathematician: what are the real lengths in an Ideal Lambert quadrilateral?

At the moment it is to hot for real mathematics but I wanted to have a function that relates the lengths of the real sides of an Ideal Lambert quadrilateral An Ideal Lambert quadrilateral (my term, ...
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1answer
46 views

How to find the value of $2g(1)+2f(1)-h(1)$?

If $$\lim_{ m\to\infty }{ \frac { x^{ m }f(1)+h(x)+1 }{ 2x^m+3x+3 } }$$ is continuous at $x=1$ and $g(1)=\lim_{ x\to0}(\ln x)^{ 2/\ln(x) }$ then how to find the value of $2g(1)+2f(1)-h(1)$? Assume ...
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1answer
27 views

$f(x)=y$ while $g(y)=x$; Is it possible to find two not reverse functions that behave such at least for a given set of inputs and outputs?

I want to know if it is possible to program such a code that could determine two distinguish, not inverse, functions, say $f$ and $g$, that is true for the below statements at a given input and output ...
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0answers
22 views

Please write down exact output after the following statements are proceesed [on hold]

Please write down exact output after the following statements are proceesed ...
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2answers
16 views

Invalid function or invalid domain

Let $ f : A \rightarrow B $ What happens if $\exists\ a\in A $ which doesn't map to any element in B ?
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0answers
8 views

Value distribution into random variables

I want to distribute a large number into quantities of 100s, 50s, 25s, 5s, 1s Lets say, I'm selling apples in quantities of 100s, 50s .... 1. Based on the quantity there is different pricing. 1 apple ...
0
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1answer
27 views

How to calculate powers of a permutation in cyclic notation? [on hold]

How do I calculate powers of an 8-cycle (1 2 3 4 5 6 7 8) ?
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2answers
9 views

Is the sum of a unimodal and increasing function still unimodal?

There is no specific function. I would like to know if there is information on summation of a unimodal and increasing function? (Continuous functions)
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2answers
46 views

An injection from R × {0, 1} to R [on hold]

What would be an example of this An injection from R × {0, 1} to R i think it is all real numbers f(x) = x Can some one help me on this. Thanks in advance
1
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0answers
10 views

A property of Quasiconvex functions.

Let f be a strictly quasiconvex differentiable function and Df denote its gradient. Is the following implication true? :"Whenever f(y) < f(x), we also have (Df(x))'(y - x) < 0" . Suppose that f ...
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0answers
19 views

Is there a function of, say, x and y that would take the first x factors in a factorial and return a xCy amounts of terms with y factors in each term?

What I'm basically looking for is described in the title. Here are some examples of what the function I'm looking for should do. Is there an existing function that does this? Even if not, are there ...
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0answers
9 views

how to show a concave function on discrete domain increases in x?

Define $g(x)=\frac{f(x)/x}{f(x)-f(x-1)}$ where x$\in$ $\mathbb{Z}$. Known that $f(x)$ has the concave extension in every consecutive $x$, i.e: $f(x+1)+f(x-1)-2f(x)<0$ holds $\forall x$. My question ...
-1
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0answers
21 views

Consider the integral expression in $x$ [on hold]

$P(x)=x^3+x^3+ax+1$ where $a$ is a rational number. At $a=3$ the value $P(x)$ is a rational number for any $x$ which satisfies the equation $x^2+2x-2=0$, and in this case the value of $P$ is $12$.
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1answer
71 views

If $f(f(x))=x$ does that mean $f(x)$ equals its inverse?

Given any real function, if $f(f(x))=x$ does that mean $f(x)$ is its own inverse? I am confused since $f^{-1}(f(x))=x$ and this is a fact, so can we assume that $f(x)$ will equal $f^{-1}(x)$ by ...
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2answers
29 views

Multivariable function as a set of functions

Consider a function $f:\mathbb{R}^n \to \mathbb{R}^m$. I've understood that it can be seen as: $f_i = (f_1,f_2,\ldots ,f_m)$, where $f_i: \mathbb{R}^n\to \mathbb{R}$. What are $f_i$ exactly? ...
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0answers
11 views

Existence of subgradient of a quasiconvex function

Does a continuous quasiconvex function always have a subgradient? More strongly, is it true that if $f$ is a continuous quasiconvex function, then for each $x$ there is a vector $c$, such that for ...
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1answer
22 views

Category of Sets and Bag-valued functions

I asked here about the Category of sets and set-valued functions, and it turns out it to be equal to REL (Category of sets and Relations),so a good studding point to study that category. Now, It ...
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0answers
27 views

Function that maps all vectors to the origin?

I need a function that will map any vector (and any point on that vector) in the cartesian plane to (0,0) using only addition and subtraction. Is this possible?
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0answers
13 views

Functions to manipulate (increase) probability exponentially or logaritmically?

Very simple. I want a function to manipulate a probability in order increase it without getting out of the range of 0 to 1. Basically a function similar to the blue lines in the following sketch: ...
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1answer
32 views

Functions of modulus

How do I calculate the range of any modulus function? I know that if $x <2$ then it's expansion is negative and if $x>2$, it's expansion is negative, but will it help? Consider an example, $$f ...
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4answers
47 views

Prove that $f(x)=m$ has three distinct real roots for $m\in(0,8)$

We have $f:\mathbb{R}\rightarrow\mathbb{R},f(x)=x^5-5x+4$ and we need to show that $\forall m\in(0,8)$, $f(x)=m$ has three distinct real roots. How can I prove it?
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2answers
52 views

What is n (Natural number) if the function has to have a limit not equal to zero or infinite? [on hold]

$$\lim_{x \to 0} \frac{(\tan(x))^n - x^n}{x^6}$$ What is n (Natural number) if the function has to have a limit not equal to zero or infinite?
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0answers
20 views

Mathematical formal expression of find “subfunction” in function

Imagine if I have a function $s(t)$ and $r(t)$. $s(t)$ may contain $r(t)$ one or more times as $s(t)$ is a quasi-period function. What is the correct expression if I want to say the $s(t)$ contains ...
3
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8answers
96 views

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$ I don't know how to find all possible values. I tried making the lower of the two roots as ...
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2answers
47 views

Is $(g \circ f)^{-1}$ equivalent to $(f^{-1} \circ g^{-1})$? Why? [on hold]

Is the inverse of composition of function is equivalent to composition of inverse functions? Why?
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1answer
20 views

Value of the difference if functions

Is there a relatively simple function $f(x)$ such that $f(x)-f(x-1)=x^n$? Note that $n$ is a positive integer. Thanks so much!
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3answers
50 views

Slopes of inverse functions

I have a question that states if $f(x) = x^3+3x-1$ from $(-\infty,\infty)$ calculate $g'(3)$using the formula $$ g'(x)= \left(\frac1{f'(g(x))}\right )$$ If I am thinking about this correctly does ...
2
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2answers
25 views

Choosing the right sign for inverse functions?

If I have to find an inverse function and through the algebra I get a $\pm$ sign how do I know which one to choose from if its in a given interval? For example a question asks: The function ...
-1
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1answer
48 views

stuck on an function question [on hold]

I'm studying CompSci, While I'm having fun with that, I haven't had a higher level math class since 2009. Unfortunately, I'm required to take calculus in order to pursue something I'm passionate ...
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0answers
31 views

One-to-One functions help [duplicate]

The one-to-one functions $g$ and $h$ are defined as follows: $$g=\{(9,8), (5,9), (8,-9), (9,-2)\}$$ $$h(x)=4x-9$$ SOLVE $$(g^{-1})(-9)=?$$ $$(h^{-1})(x)=?$$ $$(H o H^{-1})(7)=?$$
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1answer
53 views

Particular $f \in C_c^1$

is there a way to construct a function $f \in C_c^1( B(0,\frac{3R}{4}))$ such that $f|_{B(0,\frac{R}{2})}=1, \quad f|_{B(0,\frac{3R}{4})^C}=0$ and $0\le f\le 1$ everywhere such that $|Df(x)| \le ...
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0answers
52 views

Continuous function $0$ on one closed set and $1$ on the other

Looking for a better approach of the following question if possible. Question: Let $A$ and $B$ be disjoint nonempty closed sets in a metric spaces $X$, and define ...
0
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0answers
7 views

Creating an evenly distributed function $B=B(p)$ over the range $p=p_{min}$ to $p=p_{max}$

In some notes on statistical thermodynamics, I encountered this: The momentum distribution function $B(p)$ is evenly distributed over the allowed range: ...
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0answers
31 views

What is the opposite of a derangement?

A derangement is a bijection $f : A \rightarrow A$ such that $f(x) \ne x$ for all $x \in A$. Is there a name for a bijection $f : A \rightarrow A$ that is not a derangement? That is, is there a name ...
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1answer
20 views

Differentiating composite function

Can anyone say the basic formula for the differentiation of the composite functions? Is it similar to chain rule?
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1answer
48 views

How can I prove that this function is continuous in (0,0)? [on hold]

I have this function: $$ \lim_{(x,y)\to (0,0)} = \frac{2(1-\cos(xy))+\arctan(x^4)-x^2(x^2+y^2)}{(x^2+y^2)^\alpha} $$ I have to find which $ \alpha$ makes the function continuous. But my first problem ...
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1answer
25 views

Find bijective correspondence between the sets

Find bijective correspondence between the set of all functions of $X$ in the set $\left\{ 0,1 \right\}$ and the power set of set $X$ and find $| 2 ^ X |$, if $| X | = n.$ My thoughts: ...
4
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1answer
67 views

Find all functions such that $\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$

Is it possible to find all functions such that $$\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$$? My teacher asked us to give examples to prove that this is not true but I was ...
0
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1answer
27 views

Need a function for the following…

I'm trying to come up with a function formula for $y$, that is a broad curve that passes through $(0,0)$ and almost crosses $(100, y)$ but never does. ( so $x < 100$) Can you please help me? ...
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2answers
29 views

How do I reverse the smooth-step equation?

I'm using the "smooth step" equation for an easing curve: $y = 3x^2 - 2x^3$ I would like to reverse this equation so that given y, I can find ...
0
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1answer
11 views

These maps from the components into a directed system are injective when the directed system maps are.

Let $I, p_{ij} : A_i \to A_j$ be a directed system of maps such that $p_{jk}\circ p_{ij} = p_{ik}$ whenever $i \leq j\leq k$ and $p_{ii} = \text{id}$. Directed means that for any $i,j \in I$ there ...
0
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1answer
29 views

What are the composite functions

f : $\mathbb{R} \to \mathbb{R}$ $$g(x)=\begin{cases} \frac1n,&x\in\Bbb Q\text{ and }x=\frac{1}n\text{ in lowest terms}\\ \sqrt{2},&x=0\ \end{cases}$$ g(x) is the inverse of f(x) determine ...
0
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1answer
34 views

Is the function invertible?

$$f(x)=\begin{cases} \frac1q,&x\in\Bbb Q\text{ and }x=\frac{p}q\text{ in lowest terms}\\ 0,&x\notin\Bbb Q\;. \end{cases}$$ Is the function $f|_D$ invertible? If so, describe its inverse ...
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2answers
38 views

What is the domain of the function

I think the subset D is 1/n where n is an element of natural numbers. Can someone help me with this, thanks in advance
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4answers
124 views

What is the inverse of $2^x$? [duplicate]

Note: This may not be correct mathematical term, so in case of confusion, I mean what division is to multiplication. If not, just poke me in the comments. I was given this the other day: $2^x=8$ ...
1
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1answer
33 views

what does this phrase mean?

Can someone explain in everyday language what "monotonically" and "quadrilaterally" mean in the following sentence? Muscular torques increase monotonically with increasing diameter for lids with ...