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

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2
votes
2answers
592 views

If $f(\frac {x} {x+1})=x^2$ then $f(x)=?$

$$f \left (\frac{x}{x+1} \right) = x^2 \implies f(x)=\;?$$ I encountered this exercise, and and don't know how to solve it. In what category of math does this belong? In what book/website I can ...
3
votes
1answer
501 views

Nested radicals

Let $S$ be the set of functions $f:\mathbb{R}\to \mathbb{R}$ such that $\sqrt{f(1)+\sqrt{f(2)+\sqrt{f(3)+\dots}}}$ converges. A function $q(x)$ dominates $p(x)$ if there exist an m such that $q(x)\gt ...
0
votes
2answers
3k views

Onto and One-to-One of a 2 variable function

Pretty easy when its only a function of the form $y = ax + b$, however I'm trying to find if this function: $f(x,y) = 2x + y$ is onto and one-to-one. Is there a procedure to follow to check this? ...
1
vote
3answers
311 views

Find lowest integer value with integer x of given function

I need to find integer x with which function's y gets lowest integer values $$f(x)=\frac{x^2-x-17}{x-2}$$ I tried to find derivative, but it never equals 0. Other steps was to change expression ...
2
votes
1answer
556 views

Stretch Logistic Function

Follow up from: http://stackoverflow.com/questions/5225061/exponential-decay-surrounding-bounding-box How can I manipulate a simple logistic function like this one: So that the lower bound is at ...
0
votes
1answer
160 views

Injective on $L^n$: what does it mean?

I came across the following sentence in an article: $f$ is injective on $L^1(\mathbb{R})$ where $f$ is some function of $g$. Now I have a vague idea that injective= one-to-one, and can visualize ...
3
votes
2answers
272 views

What is the name of the function $f(k) = \text{max}(k,0)$ on $\mathbb Z$? [duplicate]

Let $k\in \mathbb Z$; is there a name for the function $f(k)$ below? $$ f(k) = \text{max}(k, 0) $$
4
votes
2answers
528 views

The inverse of a certain tricky function

What is the explicit form of the inverse of the function $f:\mathbb{Z}^+\times\mathbb{Z}^+\rightarrow\mathbb{Z}^+$ where $$f(i,j)=\frac{(i+j-2)(i+j-1)}{2}+i?$$
2
votes
1answer
303 views

Is this parametric curve space-filling? Why or why not?

Really, the curve in question is the polar plot $ r = cos( K * \theta) $, where $K$ is any irrational number (I use $\pi$), but the transformation to a parametric one on $x$ and $y$ with domain $t$ is ...
4
votes
7answers
939 views

How do I come up with a function to count a pyramid of apples?

My algebra book has a quick practical example at the beginning of the chapter on polynomials and their functions. Unfortunately it just says "this is why polynomial functions are important" and moves ...
2
votes
1answer
570 views

Conventions for function notation

Hey there world, I'm in year 11 right now and I just had a brief discussion with my maths teacher about function notation, specifically, how to write the result of a function squared, as apposed to ...
4
votes
3answers
511 views

How to find a Newton-like approximation for that function?

I want to find the complex fixpoint $t=b^t $ for real bases $b> \eta = \exp(\exp(-1))$. added remark: I'm aware that there is a solution using branches of the Lambert-W-function, but I've no ...
1
vote
1answer
59 views

Is there an clear way to state truncation?

How can I express that if an expression evaluates to a negative number, I assign it a value of zero? Is it possible to do this without repeating the function twice? Here is an example ...
6
votes
3answers
270 views

A calculus question

On the interval $(0, \infty)$,the function $f \geq 0$,$f' \leq 0$, and $f'' \geq 0$.Prove that $\lim\limits_{x \to \infty} xf'(x) = 0$.
1
vote
4answers
211 views

Understanding $y=|mx+n|$

The diagram shows the graph of $y=|mx+n|$ (i tried my best to do the same thing as my exercise book, actually 1 is propotional to 1 and 3 is propotional to 3, but 2 is not propotional to 2) Find ...
4
votes
3answers
704 views

How to invert this exponential function to solve for x: $y = a \exp(bx) + c \exp(dx)$?

Cheers. So if I don't make sense, I have a value for $y$, I need to know what $x$ is. $$y = a \exp(bx) + c \exp(dx)$$ $a = 12.85$, $b = 0.001857$, $c = -54.24$, $d = -0.05316$
0
votes
2answers
261 views

Mathematical function to check whether its parameter is zero or not

i need a mathematical function F which is defined as F(x) = 0 if x equal to zero and F(x) = 1 if x is not equal to zero. can F(x) be represented in the form of a single mathematical expression in ...
4
votes
5answers
258 views

Function that maximizes a function

Let's say we have a real, continuous, positive function f(x) for which we define the quantity: $$\pi(f,a) = \frac{\int_0^a f(x) dx}{\int_0^a \sqrt{1+\left(\frac{df(x)}{dx} \right)^2 ...
10
votes
4answers
5k views

Create unique number from 2 numbers

is there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 and 30, 1 and 15, 4 and 60. In general, if I take 2 random numbers result must ...
1
vote
1answer
105 views

Function composition:Find $g$

Find $g$ a) If $g∘(2f)=f+h$ where $f(x)=2x + 5$ and $h(x)=x^3 -2x$ b) If $(2f)∘g=f+h$ where $f(x)=\ln(x+2)$ and $h(x)=\sin(x^2)$ Thanks
0
votes
2answers
40 views

Reversing bijections defined via conditional expressions

Let's say that I have a variable $j$ defined by the following formula: $$j=\frac{n(n+2) + m}{2}$$ where $n$ and $m$ are two parameters, both integers, that satisfy the following conditions: $n\in ...
5
votes
3answers
494 views

Proof of linear independence of $e^{at}$

Given $\left\{ a_{i}\right\} _{i=0}^{n}\subset\mathbb{R}$ which are distinct, show that $\left\{ e^{a_{i}t}\right\} \subset C^{0}\left(\mathbb{R},\mathbb{R}\right)$, form a linearly independent set ...
1
vote
2answers
192 views

Limit of function of a set of intervals labeled i_n in [0,1]

Suppose we divide the the interval $[0,1]$ into $t$ equal intervals labeled $i_1$ upto $i_t$, then we make a function $f(t,x)$ that returns $1$ if $x$ is in $i_n$ and $n$ is odd, and $0$ if $n$ is ...
5
votes
3answers
2k views

Injective and Surjective Functions

Let $f$ and $g$ be functions such that $f\colon A\to B$ and $g\colon B\to C$. Prove or disprove the following a) If $g\circ f$ is injective, then $g$ is injective Here's my proof that this ...
5
votes
5answers
3k views

Difference between kernel and function?

I have been looking around for this question, but all results I found only describe the definition and not the answer I seek. Is "kernel" basically a synonym of "function"? When should be the time we ...
3
votes
1answer
2k views

What is an envelope of a function?

What is the definition of an envelope of a function ? For example if we multiply a sinusoid function of certain frequency ($1/f$ < support of bump) with a bump function we get a function whose ...
2
votes
2answers
167 views

power of a sets

I need to figure out what is the power of the group of functions from R to R that for each x that is not from Q, it's f(x) belongs to {x,x+1} ...
2
votes
2answers
200 views

Injective maps from $B^A$ to $(C^B)^{C^A}$

Let $A$, $B$, and $C$ be nonempty sets. I need to find conditions to guarantee that there is an injective function from $B^A$, the set of all functions from $A$ to $B$, to $\displaystyle (C^B)^{C^A}$. ...
2
votes
0answers
46 views

Deriving functions for empiracal distributions -very applied mathatics

First I am a new user of this site. Second my math background is very limited, although I do have a lot of experience in applied statistics. Component or piece part failures on high value parts($1000 ...
1
vote
2answers
126 views

Calculating the formula for a graph

I am trying to work out the formula used for this graph, I don't really know where to start, does anyone have any suggestions? Its a bit unclear, its showing FAT, Mach No, CIT, FWD B/P Position. Link ...
2
votes
0answers
132 views

Proof : Given that function $f(x)$ vanishes at 0, can we rewrite $f(x) = x_1b_1(x_1)+x_2 b_2(\overline { x_2})+\cdots+x_nb_n( x)$?

Let $f(x):\mathbb R^n \to \mathbb R$ be a smooth function, and let $f(0)=0$. I alway see that someone rewrite the function in the form $f(x)=x_1b_1(x_1)+x_2b_2(\overline {x_2})+...+x_nb_n(\overline ...
3
votes
2answers
1k views

Deriving formulas for recursive functions

If I had a recursive function (f(n) = f(n-1) + 2*f(n-2) for example), how would I derive a formula to solve this? For example, with the Fibonacci sequence, Binet's ...
0
votes
1answer
192 views

Restoring the function by its graph

I need a function that will produce a graph similar to the one below. This function is odd, symmetrical relatively to origin in III quarter. A is an asymptote (the top part is similar to ...
5
votes
3answers
165 views

For any even function $f$ there is infinite number of functions $g$ such that $f(x) = g(|x|)$

This is question from Spivak's Calculus. Question statement (paraphrased): For any even function $f$ there is infinite number of functions $g$ such that $f(x) = g(|x|)$ I have made attempt at proof, ...
4
votes
2answers
132 views

How to prove $|f(x) - f(y)| < |x - y|$ if $f(x) = x + 1/x$ where $x > 1$

I have attempted as follows: $|f(x) - f(y)| = |x + 1/x - y - 1/y|$ $\leq |x - y| + |1/x - 1/y|$ Struck here. Any help.
4
votes
3answers
145 views

Is concavity of a real-valued function on a Euclidean space implied by concavity of its restriction to every lower dimensional affine subspace?

Consider a function $f$ over $\Re^n$ to $\Re$. Suppose it is true that for every affine subspace with dimension strictly lesser than $n$ the function $f$ is concave. Is the function $f$ concave over ...
5
votes
1answer
1k views

Is any divergence-free curl-free vector field necessarily constant?

I'm wondering, for no particular reason: are there differentiable vector-valued functions $\vec{f}(\vec{x})$ in three dimensions, other than the constant function $\vec{f}(\vec{x}) = \vec{C}$, that ...
8
votes
2answers
300 views

$ \sum\limits_{i=1}^{p-1} \Bigl( \Bigl\lfloor{\frac{2i^{2}}{p}\Bigr\rfloor}-2\Bigl\lfloor{\frac{i^{2}}{p}\Bigr\rfloor}\Bigr)= \frac{p-1}{2}$

I was working out some problems. This is giving me trouble. If $p$ is a prime number of the form $4n+1$ then how do i show that: $$ \sum\limits_{i=1}^{p-1} \Biggl( ...
1
vote
2answers
139 views

Can the inverse of this logit-like transformation be stated analytically?

For $\alpha \geq 0$ the transformation $x \mapsto \log(x) - \alpha \log(1-x)$ maps the unit interval to the real line (in fact for $\alpha = 0$ the transformation is not surjective). For $\alpha=1$ ...
23
votes
6answers
686 views

Is there a function with this property?

Is there a real function over the real numbers with this property $\ \sqrt{|x-y|} \leq |f(x)-f(y)|$ ? My guess is no but can anyone tell me why? This came up as a question of one of my collegues and ...
2
votes
5answers
430 views

Prove the set of functions $f : \mathbb{Q} \rightarrow \{1,2,3\}$ uncountably infinite

Prove that the set of functions $f: \mathbb{Q} \rightarrow \{1,2,3\}$ is uncountably infinite. I'm totally stuck on this one. We have just been shown Georg Cantor's diagonalization argument in class ...
1
vote
3answers
2k views

Sums and products of bounded functions

I have been on this one for hours, cant figure out how to write this in the proper format/wording. Let $f$ and $g$ be functions from $\mathbb{R}$ to $\mathbb{R}$. For the sum and product of $f$ and ...
3
votes
2answers
185 views

A subset of permutations on the set S such that the elements that are within T, a subset of S, are not exchanged with elements outside of T

Sorry about the title, I have no idea how to describe these types of problems. Problem statement: $A(S)$ is the set of 1-1 mappings of $S$ onto itself. Let $S \supset T$ and consider the subset ...
1
vote
2answers
603 views

Example of analytic piecewise-defined function

Does there exist an analytic everywhere, piecewise-defined function $f$ such that: $f(x) = g(x)$ for $x < k$ $f(x) = h(x)$ for $x>k$ $f(x) = r$ for $x=k$ With $g \ne h $ ($g$ not the same ...
2
votes
1answer
487 views

Are there functions satisfying the following integral condition?

Can we find two functions $f$ and $g$ that are reasonably defined nontrivial (not everywhere zero, $f\neq g$, not linear polynomials) functions such that the following condition is satisfied? $$ f ...
1
vote
2answers
128 views

Does the term localized function exists?

I am looking for a term that describes function that is "localized". What I mean is a function that is non zero in a bounded range and zero else where, such as the a rectangle pulse function. But ...
1
vote
2answers
171 views

Construct a function to “increment” a value while asymptotically approaching a limit

Suppose I have a value $a_1$ such that $-L < a_1 < L$. I need to compute $a_2$ from $a_1$ and $b$, while satisfying the following requirements: $-L < a_2 < L$ if $b = 0$ then $a_2 = a_1$ ...
15
votes
4answers
4k views

Is there a bijection between $(0,1)$ and $\mathbb{R}$ that preserves rationality?

While reading about cardinality, I've seen a few examples of bijections from the open unit interval $(0,1)$ to $\mathbb{R}$, one example being the function defined by $f(x)=\tan\pi(2x-1)/2$. Another ...
3
votes
1answer
147 views

How many possible solutions for 6 wires?

Imagine 2 sets of 6 wires. How would I find how many possible connections there are? Every wire must be used to be considered a connection. ...
1
vote
5answers
1k views

How can I find inverse of function?

My question is ; How can I solve the following inverse of function question? $y=x^2-6x+4$ Thanks in advance,