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

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0
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1answer
32 views

Convolution of a continuous function and uniform continuity

Suppose $f$ is a continuous function, and let the convolution $f_n(x) := f \star \varphi_n(x)$ where $\varphi_n$ are smooth test functions. We know $f_n \in C^\infty$. We know that if $f$ is ...
0
votes
2answers
33 views

How can i resolve this equation?

Consider the following property $ P(n) $: $ \sum_{k=1}^{n} k = \frac{1}{8}(2n+1)^2 $ Show that $\forall n (P(n) \Longrightarrow P(n+1))$ Where do i start?
0
votes
0answers
13 views

What is the formula for single frequency generation function obtained from FFT?

What is the correct formula of a function that generates specific tone from fourier transform? I thought that having: transformata - an array with FFT of a source sample. v = transformata[freq] - ...
0
votes
1answer
20 views

Functions (Finding Inverse)

$f(x) = x^2 + 2x$ , domain ${x ≥ 1}$ Question: find the inverse The inverse is $f(x) = 1 + \sqrt(1+x)$ (taking the positive square root only) As $f^{-1}(5) = 2$ and as 2 is an element from the ...
5
votes
2answers
121 views

Prove that no function exists such that…

The exercise goes like this: Find a continous function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall c \in \mathbb{R}$ the equation $f(x)=c$ has exactly 3 solutions; Prove that no ...
1
vote
1answer
39 views

If $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$

If a mapping $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$ The mapping $f$ is injective as $f(x) = ...
0
votes
1answer
12 views

Functions inverse + domain

Question part a): !([http://imgur.com/sKJbFKu]) Answer: !http://imgur.com/jRfeXkW Can anyone explain why the inverse must be the negative square root?
2
votes
3answers
42 views

Why is the inverse of this function not a function?

Why does $F^{-1}$ need to be defined on all of $Y$? I can have this function: $g(x)=x,\quad x\ne 3$ and even though it is not defined for all $x$ in its domain, it is still a function, right?
0
votes
1answer
22 views

Fixed Point Iterations for Bounded Affine Functions

Let $f: X \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ compact and convex. From Brouwer fixed-point theorem, $f$ admits a fixed point ($\bar{x} \in X$ such that $f(\bar{x}) = \bar{x}$). ...
1
vote
1answer
23 views

Something basic; why do I get two different bounds on $f(x) = \frac{x^2}{\sqrt{x^2 + n^{-1}}} + \sqrt{x^2 + n^{-1}}$?

Let $n$ be a natural number. Let $f(x) = \frac{x^2}{\sqrt{x^2 + n^{-1}}} + \sqrt{x^2 + n^{-1}}$. since $x^2 + n^{-1} \geq x^2$, it follows that $$|f(x)| \leq \frac{x^2}{|x|} + \sqrt{x^2 + ...
0
votes
2answers
51 views

Please advise on the order of calculation

I have to make a software making the calculations below over some set of data. That is basically not a problem. The problem I have is with notation of the second formula which is the (Utility ...
0
votes
1answer
86 views

Inverse of a function

From my text book it says that $f(x)= x^3$ and $f^{-1}(x) = \sqrt[3]{x}$ , which I totally agree with. why does $f(x)= \frac 1 {x-1}$ and $f^{-1}(x)= \frac 1 {x + 1}$ and not equal $f^{-1}(x)= \frac ...
1
vote
2answers
38 views

Construct a specific function for a given sequence

Given the sequence $A_n=n$, I want to construct a function $f : R \to R$, such that for every $x \in N$: $f(x)=A_x$ $\int\limits_{0}^{x}f(t)dt=\sum\limits_{i=0}^{x}A_i$ How can do that? Thanks
2
votes
0answers
23 views

Intuition - Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5] [duplicate]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection ...
2
votes
1answer
81 views

$\textbf{Q}$ fails to prove some correct $\forall$-rudimentary sentence

Show that the existence of a semirecursive set that is not recursive implies that any consistent, axiomatizable extension of Q fails to prove some correct $\forall$-rudimentary sentence. I ...
1
vote
1answer
21 views

Single variable function derivative w.r.t. time?

I was studying calculus and I had doubts about this problem: (this is not homework) A circular wire expands due to heat so that its radius increases with a speed of $0.01 ms^{-1}$. How rapidly does ...
1
vote
1answer
28 views

How to establish $\sum_{d|n}d\phi(d)$

I am focusing on #5(b). I do not understand how they go from what I have to the answer. Those are r's at the end.
0
votes
3answers
31 views

invertibility of $f^{-1}$

In my introductory maths book there is a statement (it follows a theorem) that says:"Note that if f is one-to-one, then $(f^{-1})^{-1} = f$, and so $f^{-1}$ is invertible and also one-to-one because ...
0
votes
1answer
40 views

Before real numbers are precisely defined, $f(x+y) = f(x)+f(y)$ and $f(xy) = f(x)f(y)$… show $f$ preserves order.

Spivak Calculus, 4th ed., problem 3-17: If $f(x)=0$ for all $x$, then $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x$ and $y$, and also $f(x\cdot y ) =f(x)\cdot f(y)$ for all $x$ and $y$. Now suppose ...
1
vote
3answers
39 views

Functions definition + question

Am I correct in saying that for Functions, the below is the correct definition: For each value of x in the domain there is only one value of y in the range. Hence, the picture below means that it is ...
0
votes
1answer
33 views

Showing a function is decreasing

I have $$a_{n} = \left|\int^{(n+1)\pi}_{n\pi} x^{-p}\sin{(x)}~\mathrm{d}x\right|$$ and want to show this is monotonically decreasing, how would I do this? Note $n\in\mathbb{N}$ and $p>0$.
0
votes
1answer
40 views

How can a function not be one to one and be a function?

My understanding of the definition of a function Given any x, there is only one y that can be paired with x My understanding of a 1 to 1 function Given any y, there is only one x that can be paired ...
2
votes
9answers
5k views

Example of functions that are onto but not one-to-one

I have been preparing for my exam tomorrow and I just can't think of a function that is onto but not one-to-one. I know an absolute function isn't one-to-one or onto. And an example of a one-to-one ...
0
votes
2answers
38 views

Range for the function $f(x) = 3x + 2$ with domain $x > 0$

The function below is defined for continuous domains Sketch the graph and state the range of the function Question: $f(x) = 3x + 2$ for the domain $\{x \in \mathbb{R} : x > 0 \}$ The straight ...
1
vote
2answers
60 views

Problem : Solve $|x^2+x-4| =|x^2-4| +|x|$

Problem : Solve $|x^2+x-4| =|x^2-4| +|x|$ We can find the critical point of each modulus function individually then we get : $x =\pm 2;$ and $x = 0$ $x = \frac{-1 \pm \sqrt{17}}{2}$ So there are ...
-1
votes
1answer
47 views

Equating coefficients

Excuse me,i don't know how to deal with this problem,i try it for all time of last night, this equation is on "Concrete Mathematics" page 200: d(n) is the number of derangements. e^z is the ...
1
vote
4answers
78 views

$f[A]\cap f[B]\supsetneq f[A\cap B]$ - Where does the string of equivalences fail ? [Chartrand 3E 9.12(b), 9.29]

I only realised that equality may fail in $f[A]\cap f[B]\supseteq f[A\cap B]$ (i.e., that we can have $A,B,f$ for which $f[A]\cap f[B]\neq f[A\cap B]$) after checking the answer. I don't see any ...
1
vote
1answer
24 views

Roots of Taylor's series.

Show that there is exactly one value of x which satisfies the equation $$2cos^2 (x^3+x)=2^x+2^{-x} $$ I solved this using Taylor's series: $$2^x+2^{-x}=2\{1+\frac {x^2 \{ln2\}^2}{2!}+\frac {x^4 ...
0
votes
0answers
31 views

HELP! Marginal Probability Density Function of X and Y (pictures).

I understand the formulas for finding the marginal PDF of X and Y, however, in this example, how do we get from this to that: Thanks a lot!
7
votes
4answers
273 views

Is this proof correct for : Does $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$?

Is this proof correct? To prove $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$. Let any number $y\in F(A)\cap F(B)$. We want to show $y\in F(A\cap B).$ Therefore, $y\in F(A)$ and ...
-1
votes
0answers
40 views

Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5] [closed]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection $h:A\rightarrow ...
3
votes
1answer
40 views

Function in $H^1$, but not continuous

By the Sobolev embedding theorem, if $\Omega$ is bounded, $H^s(\Omega)\subset C(\Omega)$ for $s>1$, in $\mathbb R^2$. Where I can find a counterexample (if one exists) for the case $s=1$? I mean a ...
0
votes
2answers
53 views

How do I prove this bijection?

The number of $n$-digit binary numbers with exactly $k$ $1$s equals the number of $k$-subsets of $[n]$. I think i'm on the right track, but I'm confused on how to write how it's onto and 1-1. This ...
0
votes
0answers
10 views

Discrete Math Trace recursive function

Does anybody know how to trace this function by specifying the recursive calls to the function? The inputs are: A = {24, 15, 7, 10, 8, 30}, i = 2, n = 6 RandomElement(A) returns an element of A ...
3
votes
3answers
105 views

Solve $6^x=36\cdot (9.75)^{x-2}$

I try to solve the following equation: $6^x=36\cdot (9.75)^{x-2}$ I tried: $1=6\cdot (9.75)^{x-2}$ But this is obviously wrong! I think it would be smarter to bring the whole expression on one ...
2
votes
1answer
40 views

What is the use of iterating over a function?

If we have a function, say: $$ f(x) = 3x $$ We can get output values based on linearly increasing input: $$ f(1) = 3(1) = 3 $$ $$ f(2) = 3(2) = 6 $$ $$ f(3) = 3(3) = 9 $$ $$ ... $$ Or, we can ...
5
votes
2answers
55 views

Problem solve functional equation

Solve functional equation:find all strictly monotone functions $f:(0,+\infty)\to(0,+\infty)$ such that $$(x+1)f(\dfrac{y}{f(x)})=f(x+y),\forall x,y>0$$
1
vote
3answers
92 views

Recurrence relations (Big-O notation)

Say I'm given a recursive function such as: function(n) { if (n <= 1) return; int i; for(i = 0; i < n; i++) { function(0.8n) } } ...
1
vote
1answer
18 views

Convert sum to function

I need to convert $\sum_{i=0}^N \frac{C_1}{C_2+C_3i}$, to a function $C_1$, $C_2$ and $C_3$ are constants. I am interested in resulting function itself and method as well.
1
vote
3answers
70 views

Show that if $f \circ g$ is surjective, then $f$ is surjective, and $g$, the function applied first, needs not to be.

Show that if $f \circ g$ is surjective, then $f$ is surjective, and $g$, the function applied first, needs not to be. (Note:$f \circ g=f(g(s))$, $f$ and $g$ are well defined) This statement originates ...
0
votes
2answers
22 views

Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
4
votes
3answers
264 views

Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality

Let $f: A\longrightarrow B$ be a function. 1)Prove that for any two sets, $C,D\subseteq A$ , we have $f(C) \setminus f(D)\subseteq f(C\setminus D)$. 2)Give an example of a function $f$, and sets ...
3
votes
2answers
57 views

Determine the number of zeros in the first quadrant

This is a homework question: $$f(z) = z^2 - z + 1$$ sorry for the poor code!
1
vote
1answer
80 views

how to solve binary form $ax^2+bxy+cy^2=m$, for integer and rational $ (x,y)$

solve $ 3x^2+3xy-5y^2=55$ using number theory tools ,i have found the following $\Delta=3^2+4(5)(3)=9+60=69$ $d=69,u=1$ $w_{69}=\frac{1+\sqrt{69}}{2}$ ...
0
votes
1answer
27 views

Almost continuity implies measurability?

Trying to prove the continuity of $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ $(n>1)$ I got the following property of $ f $: for all $x\in \mathbb{R}^n $ and $(x_k)$ such that $x_k \rightarrow x$ ...
1
vote
0answers
24 views

How to show the inequation of two function on two different values

Denote $v_1(\lambda)=\frac{(1-\lambda)(2-\lambda)-1}{\sqrt{1+(1-\lambda)^2+((1-\lambda)(2-\lambda)-1)^2}}$ and $v_2(\lambda)=\frac{1-\lambda}{\sqrt{2+(1+\lambda)^2}}$. The figure shows the curve of ...
0
votes
1answer
35 views

Convergence of series of continuous bounded functions

Let $\mathcal{C}$ be the set of continuous bounded mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ be a continuous (with respect to the $\sup$ norm) ...
2
votes
2answers
63 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
1
vote
2answers
23 views

Symmetry of product of symmetric functions

Let's say we have three functions, $f(x), g(x) \text{ and }h(x)=f(x)\cdot g(x)$, all of which are defined for $x \in [-1,1]$. If $f(x) \text{ and }g(x)$ are symmetric around the y-axis, will $h(x)$ ...
2
votes
0answers
37 views

Creating a monotonic function

I have $n$ functions $f_i(x) \{i = 1 ,...,n\} $that does not preserve the monotonic mapping order. i.e. if $x_1 < x_2$, then in general, $f_i(x_1)$ is not less than $f_i(x_2)$ (for all $i = 1 ... ...