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

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0
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2answers
22 views

Functions and its powers

Given a map $\pi: A \rightarrow B$ what is the definition of $\pi^n$ where $n$ is a positive integer? For example if $\pi(a)=b$ then is $\pi^n(a)=b^n$? Ok so if $n=3$ then ...
1
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2answers
60 views

for any set X, construct and injection from X to Power set of X

this is what i think, if i assume X to be {1,2,3} then P(x) will have {{1},{1,2},{1,2,3},{1,3},{3,2},{3,2,1},etc}} so will not, to say, 1 from X map to more than one element of P(x) ? so how can i ...
2
votes
3answers
29 views

addition of two differential functions is differentiable

I am stuck with proving the following statement. In fact, I am proving another assumption, and the proof of this would help me to proceed. Assume that $f_1$ and $f_2$ are differentiable on the ...
-1
votes
0answers
20 views

Absolute and relative maximum and minimum

1) $f(x)= 4x^4-17x^2+4$ The critical numbers I got are $\pm\sqrt{\frac{17}{8}}$. And 0. How do I find max and min. Rel and abs? ${}{}$
1
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0answers
15 views

How to change the fundamental frequency of a sample signal?

So I am dealing with a 60Hz signal that is sampled at 1kHz. This 60Hz signal has many other harmonics (eg, 120 Hz, 180Hz..... and more). For some reason, we would like it to be 50Hz. Could we ...
0
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0answers
15 views

Finding the value of the inverse function with inverse function theorem

I am stuck by the following problem. Let $h:\Bbb R^2\rightarrow \Bbb R^2$ and $$h(x,y)= (x^2+3xy+xy^3, x^3-5y^2)$$ Let $g=h^{-1}$ near $(0,1)$. Find $Dg(0,-5)$ I showed that the inverse function ...
0
votes
2answers
23 views

How to write $y=4x-x^{2}$ as a function with respect to $y$?

Can someone please help me write $y=4x-x^{2}$ as a function with respect to $y$? I need it to determine the volume of solid of revolution about the $y$ axis.
1
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1answer
18 views

Derivative relation between two equal functions

I am stuck with the following problem. Suppose $g: \Bbb R\rightarrow\Bbb R$ is $C^1$. $f(x,y)=g(x^2+y^2)$. I need to show that $xf_y=yf_x$ My attempt was: $f_x=g_x \cdot 2x$ (1) and $f_y=g_y\cdot ...
0
votes
1answer
21 views

Differentiability of two variable function with two possibilities

There is a another question which is exactly similar to my question in this website, but I think I am still confused about that too, I couldn't get it. I would be very very very thankful if someone ...
1
vote
1answer
14 views

How can I show this equality between inverses of functions?

Let $f:X\to Y$ be a function between metric spaces $X$ and $Y$. Show that for any $B\subset Y$, $f^{-1}(B^\complement)=(f^{-1}(B))^\complement$. I was able to show that they both map to ...
0
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0answers
9 views
+50

Measure the affinity between two functions

I just want to know, what is the process of measure the affinity of two functions. I have no idea about if it has a proper name at all, so bear with me. For example, given two functions f(x) and ...
0
votes
1answer
21 views

easy calculus result about images of set under a function

PROBLEM: Let $f: X \to Y $ be a function and $\{ A_{\alpha} \}_{\alpha \in \Gamma}$ be a collection of subsets of $X$, then it occurs that $$ f( \bigcap_{\alpha \in \Gamma} A_{\alpha} ) ...
0
votes
2answers
37 views

Homeomorphism: Subspace to the usual topology

So, this is in reference to a question I asked earlier. "I have to show that the following function f:(0,1)→ℝ. I will use this function: $f(x)=\frac{1}{x}+\frac{1}{x−1}$." I've figured that this is ...
0
votes
1answer
19 views

Minimizing an open box (Calc I)

A rectangular container with an open top is to have a volume of $12 \;\text{m}^3$. The length of its base is twice the width. Material for the base costs (in dollars) 10/$\text{m}^2$. Material for ...
0
votes
0answers
36 views

Requesting information on constructed discontinuous functions (from any perspective)

Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. Define the function $F:\mathbb{R}\rightarrow \mathbb{R}$ as $$F(x)=f(x)\prod_{n=1}^\infty\frac{x-\frac{1}{n}}{x-\frac{1}{n}}$$ I ...
0
votes
1answer
29 views

Function Relations

Suppose a function $f : A → B$ is given. Define a relation $∼$ on $A$ as follows: $a_1 ∼ a_2 ⇔ f(a_1) = f(a_2)$. a) Prove that $∼$ is an equivalence relation on $A$ b) Since $∼$ is an equivalence ...
0
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1answer
31 views

How do I parametrise this expression

I have no idea how to deal with the Mins when attempting to parametrise this. How do I do it?
0
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2answers
15 views

2 and/or 4 belongs to f(x) image?

$$f(x) = \frac{x}{x^ 2+x}+4$$ How can I now if 2 and or 4 belongs to the image of the function? (in a General Way so I can apply it to other functions)
-1
votes
0answers
24 views

$f(n - k) = \theta(f(n))$, for positive constant $ k > 1$ [closed]

Prove/disprove: There is exist function $f(n)$, s.t. $f(n - k) = \theta(f(n))$, for positive constant $k > 1$. thanks all.
1
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5answers
83 views

Let A be a set of all infinite sequences consisting of 0's and 1's. Prove that A is not countable.

Sequences such as {010101010101...., 10100100100...., etc} if i am not wrong these sequences can represent all the real numbers in the binary format, so a such a set will not be countable. but i am ...
2
votes
4answers
93 views

Show bijection from (0,1) to R

I have to show that the following function $f: (0,1) \rightarrow \mathbb{R}$. I will use this function: $f(x)=\frac{1}{x}+\frac{1}{x-1}$. To show 1-1, I am using $f(x_1)=f(x_2) \Rightarrow x_1=x_2$, ...
1
vote
1answer
26 views

Identifying sets

I keep seeing written in texts the phrase 'identify sets', for example: Identify $A$ as a subset of $F(A)$ by $a\mapsto f_a$, where $f_a$ is the function which is 1 at a and 0 elsewhere. This is in ...
0
votes
1answer
17 views

Prove or disprove: functions

For a subset $X$ of $Y$, $f^{-1}(X') = (f^{-1}(X))'$ Is this true? If so, what would be a proof for it? If it isn't, what would be a counterexample? I'm completely lost The $'$ denotes "not" or the ...
2
votes
1answer
30 views

Why is $x$ restricted this way? (limits of functions)

Here is a corollary from Ross' Elementary Analysis: Why is $x$ restricted this way?
0
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0answers
12 views

Help with understanding why a particular set was chosen in theorem and corollary involving the limit of a function

Here's a theorem from Ross' Elementary Analysis, he gives the limit definition of a function in terms of $\delta$-$\epsilon$ And the corollary that follows: Why is it that in the corollary, the ...
3
votes
1answer
28 views

Contraction vs. iterated function convergence

Let $X$ be a Banach space with norm $|x|_X$. (For example $X=\mathbb R$) We assume that a function $F: X \rightarrow X$ is Lipschitz continuous but not a contraction map, hence: $$ | F(x) - F(y) |_X ...
0
votes
0answers
6 views

Sum of values of a function with concurrent lines

Lets say I have a linear function as follows: $y=k*x_0$, where $k$ is NOT a constant. Values of $k$ can be between $0$ and $k_1$. In this example values of $k$ range from $k_2$ to $k_3$. My ...
0
votes
0answers
16 views

Function Notations

Let $f:X\to Y$ be a function and let $x\in X$. For the image of $x$ under $f$, a popular notation is $f(x)$; while some author prefers $(x)f$. To me, $(x)f$ is far more natural as if we apply a ...
0
votes
1answer
31 views

Approximating the definition of inverse function

Let $f:D\mapsto\mathbb{R}^n$, $D\subset\mathbb{R}^m$ open, $m<n$ be a continuously differentiable function. I define $g:\mathbb{R}^n\mapsto D$ by $$ g(y)=\operatorname{argmin}_{x\in D}\|f(x)-y\|, ...
2
votes
0answers
39 views

Help in deriving a formula

Background I am working on a vocabulary building application under which I am trying to build an adaptive test for the student. The test would be adaptive to the user's response: When the student ...
1
vote
1answer
39 views

Basic question about math injectivity

Suppose that $A:\mathbb{R}^3\to \mathbb{R}^2$ and $B:\mathbb{R}^2\to \mathbb{R}^3$. Can we guarantee that either $A\circ B$ or $B\circ A$ is injective? Is it possible for either $A\circ B$ or $B\circ ...
1
vote
1answer
74 views

How to find the price elasticity of demand?

I need help answering if this is demand elastic of inelastic. A policy adviser suggests that in order to improve its balance of trade with china, Canada should lower the price of some heavy ...
0
votes
1answer
16 views

Showing a function is a bijection without a specific function

Let $F$ be an ordered field with identities $0$ and $1'$, and define $f: \mathbb{N} \to F$ by: $f(1) = 1'$, $f(x + 1) = f(x) + 1'$ (the addition for the right hand-side is addition in the field). Let ...
0
votes
1answer
37 views

Prove $f(a + b) = f(a) + f(b)$ in an ordered field

Let $F$ be an ordered field with identities $0$ and $1'$, and define $f: \mathbb{N} \to F$ by: $f(1) = 1'$, $f(x + 1) = f(x) + 1'$ (the addition for the right hand-side is addition in the field). So ...
1
vote
1answer
16 views

Continous function with one of ranges as an equation

I'm pretty new here and my formatting might have some errors, sorry I could get it only this far. f(x) = \begin{cases} \ ax+2b, & x<0 \\[3ex] \ x^2 + 3a -b, & x^2 + 3a -b \\ \ 3x-5, ...
1
vote
1answer
208 views

The image of a map $H^2 \setminus \Delta \to \mathbf{R}^4$ where $H$ is the upper half-plane and $\Delta$ is the diagonal

Let $H = \{z \in \mathbf{C}: \operatorname{Im} z > 0\}$ be the upper half-plane, and let $D = H^2 \setminus \Delta$, where $\Delta = \{(u,u) \in H^2\}$ is the diagonal. Define $\varphi: D \to ...
0
votes
2answers
19 views

Extension of a function to a continuous function

The problem is the following: Extend the following function to a continuous function defined on all $\mathbb R^2$ $f(x,y)=xy/(x^2+y^2) $ I have never solved such a problem. Would be thankful if ...
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votes
0answers
38 views

[Find the root of the equation, please help] [closed]

I have a problem need your help, I have the following equation$$x\ln(x)+ax^2 + bx + c = 0.$$ It look like Lambert function, but I cannot find the root of this equation. Everybody can help me to find ...
2
votes
3answers
30 views

Proving $f(n + 1) > f(n)$ and is f injective?

If I have a function $f:\mathbb N \to \mathbb N$ defined for every $n \in \mathbb N$ by: $$f(n) = (n+1)!-1$$ How would I prove that $f(n+1) > f(n)$ for every $n\in\mathbb{N}$? Would it be ...
0
votes
2answers
36 views

How to create a scale operation which produce result in a specific range?

First of all, I'm not sure what what to call what I would like help creating so please bear with me. I want to create a system with 2 inputs x and y, f(xy). In which both x and y are in range. For ...
0
votes
1answer
13 views

How to Simplify this question?

$$\begin{align}f(n+1) &= (n+2)! -1 = (n+2)(n+1)! - 1 \\ &= (n+2)\left((n+1)!-1\right) + (n+2) - 1 \\ &= (n+2) \cdot f(n) + (n+1) \end{align}$$ I understand the first line but not how to ...
1
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0answers
4 views

Define a piecewise function in wxMaxima with variable pieces

I have to define a piecewise function in wxMaxima, but the number of pieces is given by a variable (n) and the interval is given by a list (list_interval). For example, $f(x)= x^2 , \ ...
2
votes
1answer
47 views

Catch 22 situation involving inverting a function and finding the range of the function.

Let $f(x) = \sqrt{x+5} - \sqrt{x-5}$ Calculating the inverse: $y = \sqrt{x+5} - \sqrt{x-5}$ $y + \sqrt{x-5} = \sqrt{x+5}$ $y^2 + x - 5 + 2y\sqrt{x-5} = x + 5$ $\frac{(10 - y^2)^2}{4y^2} + 5 = x$ ...
0
votes
3answers
62 views

How to prove that the function $f(x) = 2 \left \lfloor x \right \rfloor - x$ is one to one for rational $x$?

How to prove that the function $f(x) = 2 \left \lfloor x \right \rfloor - x$ is one to one for rational $x$? I believe that I will have to somehow use the fact that the $\left \lfloor x \right ...
1
vote
2answers
61 views

Find a formula to fit a table of values. [closed]

how do i make a formula using this input=output(rounded to a whole number) 0 =10 .1=104 .2=253 .3=489 .4=863 .5=1457 .6=2400 .7=3895 .8=6267 .9=10030 1 =16000 I have a circuit board that does this but ...
3
votes
1answer
43 views

A sufficient condition to ensure a function to be linear

Suppose that $f$ is continuously differentiable on $\Bbb R$, and $$\lim_{x\to +\infty}f'(x)$$ exists and is finite. Furthermore, $$f(x+1)-f(x)=f'(x),\ \forall\ x\in\Bbb R.$$ Show that $f$ is linear, ...
3
votes
3answers
132 views

Finding the zeroes in a function

I have come across a problem on my trigonometry homework where we need to find the zeros of a function without the use of a calculator. The Equation: Given one of the zeros is $x=5$ $f(x) = ...
0
votes
2answers
15 views

Points of Inflection

Does $f(x)=|x^2−1|$ have an inflection point at $x=(-1)$ and $x=1$? It seems like the concavity is changing, but I just wanted to make sure.
0
votes
1answer
30 views

Function Of Any Line?

If I were to scribble a line of varying curves into a sheet of paper and for each value of X there was only a single value of Y, how can I go about finding the function for such a line in a way that ...
1
vote
1answer
35 views

Differentiation map and the Cayley-Hamilton theorem

I have computed (a) to be $-\lambda^3$. I also know that the Cayley-Hamilton theorem states that substituting the matrix A (where A is matrix with p(λ)=det(λI-A) for λ in this polynomial results in ...