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

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0
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3answers
25 views

If graphs of the functions $y=ln x$ and $y=ax$ intersect at exactly 2 points then a must be?

If graphs of the functions $y=ln x$ and $y=ax$ intersect at exactly 2 points then a must be? Ok I know the graphs of these two functions but what should be the calculus based technique to find the ...
0
votes
1answer
25 views

How to prove the two functions are inverse to each other? [on hold]

How to prove the definition of following inverse function: Let A and B be two sets. Prove that: $f^{-1} : B \rightarrow A $ is an inverse function of $f : A \rightarrow B$ . (in general)
1
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3answers
13 views

Finding range of $||x| - |y||$ for the given conditions.

If $ z = x + iy$ and $ x^2 + y^2 = 16 $ , then the range of $||x|-|y||$ is...? This is what I've tried yet: Suppose $x = a\cos \theta$ and $y = b\sin \theta$, then we've : $$\begin{align} ...
1
vote
1answer
30 views

Find range of the given function : $ f(x) = \frac{e^x}{1+ [x] } $ when $ x \ge 0 $

Find the range for $ f(x) = \cfrac{e^x}{1+[x] } $ when $x\ge 0$ . Where $ [.] $ denotes greatest integer function. My book answers it in a very straight forward manner - Here f(x) is ...
0
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1answer
31 views

Suppose that a particle is located at the origin $(s = 0)$ at time $t = 1$ and moves along the line with velocity $v(t) = t^{-2}$.

Suppose that a particle is located at the origin $(s = 0)$ at time $t = 1$ and moves along the line with velocity $v(t) = t^{-2}$. How can I find the position s as a function of time? And how can I ...
0
votes
1answer
57 views

Proper notation for the function $g(x) = x^2+6$.

I'm using this more as a method of verifying if I'm correct on a question I am having difficulty with. Keep in mind, I'm a complete beginner, so.. yeah. Thereom: Assume the function $g$ is ...
0
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0answers
12 views

Function whose fixed points are a convergent sequence with derivatives at each term $\neq 1$ and derivative at limit point $=1$

The question asks to find an example of a function where the fixed points of the function are a sequence that converges to a fixed point, where the derivative of the function at each of the fixed ...
2
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0answers
26 views

Area & order-preserving function transformations

Consider a bounded function $f(x \in \mathbf{R}) \rightarrow \mathbf{R}$ with bounded support $\left[0,L\right]$ (illustration below). What type of transformations $g(f(x))$ guarantee that: Area ...
1
vote
2answers
55 views

Find whether $f(x) = x + \cos (x)$ is one-one.

Find whether $f(x) = x + \cos x$ is one-one. I tried finding $f'$ , $f'(x) = 1 - \sin x$ . But, how is this strictly increasing? Following is the graph for $1 - \sin x$ that I plotted using ...
2
votes
1answer
40 views

Where does this function come from in this proof?

This is an excerpt taken from a proof: Let each $M_n(n\in\mathbb{N})$ be countable, Then there exists an injective function $f_n:M_n\rightarrow\mathbb{N}$. Now, set a function ...
-4
votes
0answers
23 views

Vertical Distance Problem

Determine the two points on the $x$ axis at which the vertical distance between them and the respective corresponding points on the line $x-1$ is $6$ units. I understand how the difference function ...
2
votes
1answer
20 views

Find curve that fits (min, mean, max) to (0, 0.5, 1) [on hold]

I'm trying to use the fact that $log(1) = 0$ and $log(\sqrt{e}) = 0.5$ and $log(e) = 1$ to write a map from a set of data points to a value between $0$ and $1$ such that: $f(min) = 0$, $f(mean) = ...
0
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0answers
26 views

Help with solving ODE differently [duplicate]

For which values of $T$ can we find a unique solution of the ODE $x''(t) = −x(t) $ satisfying the boundary conditions $x(0) = a_1$ and $x(T) = a_2$ for any values of $a_1$ and $a_2$ ? I can solve ...
1
vote
1answer
31 views

Whether the terms Bases function and Basis function are same or differen?

I have came accross few lines in my reading/discussion like any signal can be represented by summation of elementary Signals.these elementary signals are called as basis(sometimes read as bases?) ...
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0answers
24 views

Applications of the Mean Value Theorem in Integration [on hold]

What are some interesting applications of the average value of a function when it comes to the topic of Integration? Thanks.
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0answers
35 views

Solving trigonometric function in Pi term [on hold]

How to solve [cos(x + sin(2x))][1 + 2 cos(2x)] = 0 in term of Pi? Can you show the steps for calculation. Thanks
0
votes
2answers
27 views

Find an example of continuous but not increasing function whose inverse function doesn't satisfy the Inverse Function Theorem

I have to find an example of a function $f:[a,b]→R$ which is continuous, but not strictly increasing, such that no inverse function $f^{−1}$ satisfy the property of the Inverse Function Theorem.
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0answers
15 views

Parameterizing a linear compressor

I am hoping to build a function $f_{A,B,\alpha}(x \in \mathbf{R} ) \rightarrow y \in \mathbf{R}$ that serves as a positive signal compressor. The function acts on an input signal $x\left(t\right)$ one ...
2
votes
1answer
21 views

Is this function defined in terms of elliptic $\mathrm{K}$ integrals even?

Let $R,z > 0$ be positive real constants, and consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $$ f(v) = \frac{1}{\sqrt{(R+v)^2+z^2}}\ \mathrm{K}\!\left( \frac{4 R v}{(R+v)^2+z^2} ...
0
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0answers
28 views

Thinking Problem Involving Average Rate of Change

Here is the question: Suppose that the number of houses in a new subdivision after t months of development is modelled by $N(t)=\frac{100t^3}{100}+t^3$. where $N$ is the number of houses and $0\leq ...
1
vote
3answers
20 views

Is the function max{x,y} defined if x and y take equal values?

If x and y take the same values, will the function return a result? I am asking this as maximum means greatest of two values. So if both the values are equal, the existance of the function confuses ...
2
votes
3answers
231 views

Holomorphic function

Let $f(z)$ be a holomorphic function that maps the unit disk to the unit disk. Prove that $$|f^{(5)}(0)| \leq 120.$$ I use some concrete example it seems that this statement work out but i ...
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0answers
36 views

Set of natural numbers and its functions. [closed]

Let $n \ge 1$ and S = { 1,2,....n}. For a function f: S $\to$ S, set D subset of S is said to be invariant under f, if f(x) is an element of D for all x in D. Note that the empty set and S are ...
0
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0answers
14 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 ...
-2
votes
1answer
28 views

Integration using Substitution [closed]

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 ...
2
votes
2answers
50 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 $X$, $f(A \cup B) = f(A) \cup f(B)$ ? I believe that it is true, and here is my proof. If somebody sees something I did wrong, can you please ...
0
votes
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
votes
1answer
25 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
40 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
77 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
votes
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
6
votes
2answers
471 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
36 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 ...
2
votes
5answers
101 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
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0answers
64 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
30 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 ...
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0answers
16 views

Find preimage of a function given that the image of the function [closed]

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
31 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 ...
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0answers
26 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 ...
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2answers
40 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
3answers
43 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
29 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
37 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 ...
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0answers
39 views

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

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
votes
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
28 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
30 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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votes
1answer
27 views

What is the function between x and y? [closed]

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
35 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 ...