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

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0
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0answers
7 views

Intersection of curves and constructing a plane

Can someone please help me with how to approach/solve this question? Show that the following pair of curves intersect, and construct a plane that is tangent to both curves at the point of ...
-1
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1answer
18 views

Integration using Substitution

Firstly, I know that the graph of function, $f$ must cut the x-axis at least once such that the definite integral will equal to zero so I can apply Roelle's theorem somewhere. For b (i), letting $u ...
0
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0answers
16 views

Efficiently compute the bases functions of the Signal [on hold]

Let there be 4 1D signals such that \begin{cases} x(t)=4\sin(10\pi t) \\ y(t)=8\cos(20\pi t) \\ z(t)=16\sin(30\pi t) \\ m(t)=x(t) +y(t)-z(t) \end{cases} Is there a way to compute ...
1
vote
2answers
30 views

True or false: for all subsets $A$ and $B$ of $X, f(A\cup B) = f(A) \cup f(B)\,$?

Can somebody prove For all subsets A and B of X, f(AUB) = f(A) U f(B) ? I believe that it is true, and here is my proof. If somebody sees something I did wrong, can you please explain? Take any y∈F ...
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1answer
9 views

Finding Prime Implicants and Essential Prime Implicants for Boolean Functions

I am trying to solve a EE problem and am unsure whether I doing it correctly. The problem is: Find all the prime implicants for the following Boolean functions, and determine which are essential: ...
2
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1answer
22 views

Questions concerning elements in $F = \big\{f: \{1, 2, 3\} \to \{1, 2, 3, 4, 5\}\big\}$.

a) Find and simplify the number of functions $f \in F$ so that $f(1) = 4$. My attempt: there is $1$ choice for $f(1)$, and $5$ choices for $f(2)$ and $5$ choices for $f(3)$, thus $1\cdot 5\cdot 5 = ...
1
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1answer
32 views

True or false: For all subsets $ A $ and $ B $ of $ X $, if $ A \subseteq B $, then $ f[A] \subseteq f[B] $.

I am trying to determine if the following is true or false: For all subsets $ A $ and $ B $ of $ X $, if $ A \subseteq B $, then $ f[A] \subseteq f[B] $. My guess is this would be true, because ...
3
votes
2answers
66 views

Equality of a quadratic function

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ an arbitrary function and $g: \mathbb{R}\rightarrow \mathbb{R} $ a quadratic function with the following property: For any $m$ and $n$ the equation ...
0
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1answer
19 views

Find and draw the domain of function.

please could you help us. We have to find and draw the domain of function: $$f(x,y) = \frac{1}{\ln\left(y\cos (\pi x)\right)}$$ Thank you
5
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2answers
386 views

Is this a meaningful approach to primes?

Motivation: I tend to be good at recognising patterns and I saw one with factorial: $$ n! = \prod_{i=1} p_i^{J(n,p_i)} $$ where $p_i$ is the $i$'th prime and $$ J(n,i)= \sum_{S=1}^\infty [n/i^S] $$ ...
1
vote
1answer
28 views

Calculating the volume of a cylinder.

Let $V = \{(x, y, z): x^2 + y^2 ≤ 4$ and $0 \le z \le 4\}$ be a cylinder and let $P$ be the plane through $(4, 0, 2)$, $(0, 4, 2)$ and $(−4, −4, 4)$. Compute the volume of $C$ below the plane $P$. I ...
1
vote
5answers
94 views

$f(x)f(1/x)=f(x)+f(1/x)$

Find a function $f(x)$ such that: $$f(x)f(1/x)=f(x)+f(1/x)$$ with $f(4)=65$. I have tried to let $f(x)$ be a general polynomial: $$a_0+a_1x+a_2x^2+\ldots a_nx^n$$ which leaves $f(1/x)$ as: ...
1
vote
0answers
60 views

The Real solution of the equation $x^\frac13= -1$

Surely, $x=-1$ is the only solution. But, wolframalpha says there is no solution in the real field. I think this is because they transformed $x^\frac13$ to $e^{\frac13(lnx)}$. But why did they ...
0
votes
1answer
26 views

How do I make my TI-89 evaluate a recursive function?

On my TI-89 I can assign variables recursively such as: $1\to x$ returns 1 $x+1 \to x$ returns 2 $x+1 \to x$ returns 3 etc. How could I do functions the same way: $x \to f(x)$ returns Done $2\cdot ...
-4
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0answers
15 views

Find preimage of a function given that the image of the function [on hold]

I have a following question that I need help on. Let $F(s,t)=\left( s^2 \cos(t), s^2 \sin(t) \right).$ Find the $F$-preimage of $[0,1] \times [0,1].$
0
votes
1answer
22 views

Cavalieri’s Principle for calculating volume.

Let $B = \{(x, y, z): x^2 +y^2 +z^2 ≤ 4\}$ be the ball with radius $2$ in $\mathbb{R}^3$ and let $V$ be the region inside $B$ above the plane $z = 1$. Use Cavalieri’s Principle to compute the volume ...
0
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0answers
25 views

Increasing/ decreasing functions

We are given a random variable x with a pdf f(x) and F(x) is its distribution function. Let $$r(x) = \frac {xf(x)} {1-F(x)} $$ Then for $x< e^{\mu} $ and $$f(x) = \frac {e^ {1/2(\log x - \mu)^2}} ...
0
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1answer
26 views

Theorem? For every $f:\mathbb{R}\to\mathbb{R}$, for every $A \subseteq R$ where $A$ is finite, $\exists c\in\mathbb{R}:\forall x\in A:(f(x) = c)$.

Your mathematical sense problably twitched when you read the title, as a simple counterexample of the theorem is some one-to-one function. Where then, is the mistake in this proof? Let ...
1
vote
2answers
39 views

Limit problems in two variable function

How would one find the $\alpha$ and $\beta$ for which $$\frac{x^{\alpha}y^{\beta}} {\sqrt{x^2 + y^2}} \to 0$$ as $(x,y) \to (0,0)$ ? I understand the $\epsilon$-$\delta$ definition of a limit but ...
0
votes
1answer
29 views

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions :

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions : $ f(x+t,y) = f(x,y) + ty~~;~~f(x,t+y) = f(x,y) + tx~~;~~f(0,0) = K$. Then $\forall ~~x,y \in \mathbb R, ...
2
votes
1answer
22 views

How to calculate the volume of a skip bin container knowing the height of the material inside

I need to know hot to calculate the volume of a skip bin (also known as a skip container or dumpster in some areas) with varying length and width. It seems like a isosceles trapezoid when you look at ...
0
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2answers
31 views

general formula of a sequence

Need help! I'm computing for the general formula of this sequence $S = 1 + 4t^2 + 9t^4 + \ldots + n^2 t^{2n - 2}$. I tried multiplying the equation by $t^2$ then subtracting it by the original ...
0
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0answers
37 views

Need hints on how to start on this? [on hold]

Consider the function $f(x)=x^2$ Suppose we want to get error controls for $f(x)$ at $x_0 = 1$ with $ε = 0.1$. Show that $δ = 0.05$ is not sufficient, but $δ = 0.04$ is. Now try to do the same ...
0
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1answer
29 views

Can the function y=5 be injective or surjective for all x ∈ integers?

I have a practice exam and I get kind of confused about: Is the constant function y = 5 , ∀ x ∈ Z [All integers] Is this function Injective or Surjective?
0
votes
2answers
23 views

How to find the values of a at which $y$ is increasing?

I don't know how to solve this one and the question is: Find the values of a at which $y = x^3 + ax^2 + 3x + 1$. My solution is: $y'= 3x^2 + 2ax + 3$ I know that if $y' \ge 0$, $y$ should be ...
2
votes
1answer
29 views

Order the domain so that function is monotonic

Let $f : \mathbb{R} \to \mathbb{R}$ be a measurable function. Is there a bijection $b: \mathbb{R} \to \mathbb{R}$ such that $f \circ b$ is monotonic?
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1answer
26 views

What is the function between x and y? [on hold]

X --- Y 4 | 1 3 | 2 2 | 3 1 | 4 0 | 5 I hope I am wording it right, basically I want to get my y using my x.. sooo Y = X _____
1
vote
1answer
27 views

If f is continuous and strictly increasing, then the function $f^{-1}:f(I)\rightarrow I$ is continuous and strictly increasing.

Let $I \subseteq \Bbb R$ be a non-degenerate open interval, and let $f:I\rightarrow \Bbb R$ be a function. Suppose that f is strictly monotone. If f is continuous and strictly increasing (or ...
0
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2answers
56 views

Is it possible to find the value of $x$ where $e^x$ exceeds $x^{10}$ by hand?

All I managed is to "simplify" the equation $e^x=x^{10}$ to $\frac{x}{\ln{x}}=10$. Is there some way or trick to make the equation look like $x=\dots$? (Solve the equation, in other words.)
-3
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0answers
21 views

How to determine whether injective or surjective over this functions [duplicate]

G : N×N given by G = 2x+5 ∀x ϵ N H : Z×Z given by H = 10 ∀x ϵ Z I got an idea whether injective or surjective but don't know how to go through. And finally, are these functions? I think they are ...
-1
votes
1answer
21 views

How to go towards this functions and defining whether injective or surjective

G(x) : N×N given by G(x) = 2x+5 ∀x ϵ N H(x) : Z×Z given by H(x) = 10 ∀x ϵ Z I am not familiar with this notations. However, I got an idea whether injective or surjective. And finally, are ...
6
votes
5answers
45 views

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing.

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing. I do not have use derivatives, so I decided to apply the definition of being a strictly ...
3
votes
1answer
50 views

Are there only a few 'universally convergent' Taylor Series?

The taylor series for $sin(x)$, centered at any point, converges for all x. The taylor series for $e^{x}$ and $cos(x)$ do as well. Thus, taking an algebraic function of these (without division) ...
0
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3answers
61 views

Beginner level : What is the intuitive meaning and what are the steps to prove for an injective function

For the proof that a map is into, it is convenient to use the contrapositive of the definition of one-to-one, namely: $$ \forall x,y \in X, f(x) = f(y) \rightarrow x = y. $$ where the definition of ...
6
votes
2answers
124 views

What is the meaning of $\mathbb{R}\setminus\{0\}$?

This is used in many posts related to functions and googling it doesn't help. What does this mean? $\mathbb{R}$ should stand for all Real numbers.
3
votes
5answers
66 views

Number of functions for $(f(x))^2=x^2$

If $f\colon\mathbb{R}\to\mathbb{R}$ is a function such that $$(f(x))^2=x^2$$ for all $x$ , then 1) The number of such functions are? 2) How many of them are continuous? I can see 4 functions: ...
0
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0answers
27 views

proof that some expected value equal to $\theta (\log n - \log k)$

So here is the problem - Given the following equation: $(c_2\cdot \log n) - (c_1\cdot \log k)\le E(X)\le 1+ (c_1\cdot \log n) - (c_2\cdot \log k)$ When $c_2,c_1\gt0$ and also $c_1\gt c_2$ In ...
1
vote
4answers
24 views

Is the mapping $f\colon\mathbb{R}\to\mathbb{R}$ defined by $f(x)=5x^3+3$ onto?

Let $f\colon \mathbb R \to \mathbb R$ be defined by $f(x)= 5x^3+3$. Is it onto? According to me, if $y=5x^3+3$, then $x = \sqrt[3]{(y-3)/5}$ is not an element of $\mathbb R$ for all $y \in ...
1
vote
1answer
16 views

functions and the commutative property

with regard to vector spaces of functions. How do I know if the commutative property holds for a set of functions. especially if the vector space includes an infinite set. for instance, for the ...
1
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2answers
43 views

Why the continuity of $f$ is not a necessary condition?

I am quite new to functions and continuity, and now I am reading the slides regarding the intermediate value theorem, which is related to continuity of functions. While reading, I found the ...
3
votes
2answers
18 views

Increasing/Decreasing intervals of a parabola

I am being told to find the intervals on which the function is increasing or decreasing. It is a normal positive parabola with the vertex at $(3,0).$ The equation could be $y = (x-3)^2,$ but my ...
1
vote
1answer
44 views

Find $f(x)$ given $f, g$ such that $\,f(0) =2,\, g(0) =1, \, f'(x) = g(x),\, g'(x) = f(x)$.

Let $f$ and $g$ be functions satisfying: $$\begin{align} f(0) & =2\\ g(0) &=1 \\ f'(x) &= g(x) \\ g'(x) & = f(x) \end{align}$$ Find $f(x)$.
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1answer
19 views

What is $h^{-1}(L)$, for $L$ a regular language and $h$ a homomorphism?

Let $L = L((00 + 1)∗)$ and $h : \{a, b\}^* \to \{0, 1\}^*$ be defined by $h(a) = 01$ and $h(b) = 10$. What is $h^{−1}(L)$? In this context "$+$" means "$\cup$". So the language $L$ is all the ...
3
votes
1answer
62 views

Functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$

How to find all functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$ for $x> 1$?
2
votes
1answer
20 views

given the following two conditions, find $f(x,y)$

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions: $f(x+t,y)=f(x,y)+ty$; $f(x,t+y)=f(x,y)+tx$; $f(0,0)=k$; then for all $x,y \in\mathbb R$, $f(x,y)=$ a) ...
1
vote
1answer
36 views

domain of $\sqrt {\cos^{-1}(\cos x)-\lfloor x\rfloor} $

Here is my question where I got stucked. The domain of $\sqrt{\cos^{-1}(\cos x)-\lfloor x\rfloor} $ where $\lfloor \cdot\rfloor$ denotes the greatest integer function (floor function).
0
votes
0answers
9 views

Lagrange Multiplier, Boundary

In many cases when we have to optimize a function under a constraint, i.e $f(x,y)=e^{-xy}$ with constraint $x^2+4y^2 \le1$, Lagrange multipliers only help with finding the extreme values at the ...
0
votes
3answers
27 views

Functions - Inverses of graphs.

The question reads: sketch the graph of y=-3-x along with its inverse. From calculating the equation of the inverse graph, I come to y=-3-x, using the swap method. I then tried to plot both graphs ...
1
vote
1answer
23 views

Let $f\colon [a,b]\to\mathbb R$ is continuous and $G(x,t)=t(x-1)$ when $t\leq x$ and $x(t-1)$ when $t\geq x$.

Let $f\colon[a,b]\to \mathbb R$ is continuous and $$G(x,t)=\begin{cases}t(x-1)&\text{when $t\leq x$,}\\x(t-1)&\text{when $t\geq x$.}\end{cases}$$ Let $$g(x)=\int_0^1f(t)G(x,t)\,\mathrm dt.$$ ...
-2
votes
2answers
33 views

What is the function $w(x) = 4 + \sqrt[3]{x}$? [on hold]

I need to know how to go about determining if this function is even, odd, or neither