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

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5answers
78 views

if $f(x) = x-\frac{1}{x}.$ Then no. of solution of the equation $f(f(f(x))) = 1$

If $\displaystyle f(x) = x-\frac{1}{x}.$ Then no. of solution of the equation $f(f(f(x))) = 1$ $\underline{\bf{My\;\; Try}}::$ Given $\displaystyle f(x) = x-\frac{1}{x} = \frac{x^2-1}{x}.$ Now ...
0
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1answer
132 views

Is there a name for the normal CDF function $\Phi(\cdot)$?

I can't seem to find a plain English name for the CDF of the normal distribution $\Phi(x)$. However, I am aware of several other related functions that have a name, so I feel like this one should as ...
0
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2answers
66 views

A function of $f\circ g$

This is studying for my midterm. Let $f(x)=x^2/(x+1)$ and $g(x)=2x-3$ A function of $f\circ g$ is: So I begin with the equation: $$x^2/(x+1) \cdot 2x-3$$ Add one to the denominator of the second ...
2
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3answers
75 views

For $f:\mathbb R^{<0}\to\mathbb R$, $f(x)=2x^2-3$, find the values of a for which $f(a)=f^{-1}(a)$

Okay, i've got the answer for this with some luck I guess, however i'm still left wondering specifically what this part of the question means: "find the values of a for which $f(a)=f^{-1}(a)$" My ...
2
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3answers
79 views

Equation of a line parallel to $5x-3y=7$ That goes through the point (3,-1)

This is a study question in preparation for my midterm. It's multiple choice. The answers are: A) $y=(5/3)x-(7/3)$ B) $y=(3/5)x-(14/5)$ C) $y=(5/3)x-6$ D) $y=-(3/5)x+(4/5)$ Here is my process: ...
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3answers
81 views

How to prove an equivalent definition of injective? [duplicate]

Let $A,B$ be non-empty sets and $f:A\to B$ a function. Proof that $f$ is injective, iff $f\circ g=f\circ h$ implies that $g=h$ for all functions $g,h:Y\to A$, for every set $Y$? I can see why ...
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1answer
308 views

Injectivity and Surjectivity of a piecewise defined function

Let there be a function $f\colon\def\R{\mathbb R}\R \to \R$ given by $$ f(x)= \begin{cases} 5x+2 &x\ge 1 \\ x-1 &x<1 \end{cases}, \qquad x \in \R. $$ Prove that $f$ is not surjective, also ...
2
votes
0answers
115 views

Does ternary operations have associative property?

Binary Operation is a function. Right? We know that all Binary operations have associative property. They must be either associative or non-associative. The condition is : $$(a*b)*c = a*(b*c)$$ ...
0
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1answer
149 views

Pascal's Triangle Problem?

Guys I have been trying to solve this problem for a long time but cant seem to come up with anything. Problem is essentially a Pascal triangle but I cant figure out how to sum up a column for a ...
2
votes
2answers
109 views

Find the values of $a,b,c,d$ for the equation $f(x)=ax^3+bx^2+cx+d$

How can I find the values of $a,b,c,d$ for the equation $f(x)=ax^3+bx^2+cx+d$ ? There are $4$ points given: $(-1,0)$, $(0,1)$, $(2,0)$, $(1,2)$. Thanks!
0
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2answers
481 views

Finding Taylor's expansion for $f(x) = \sqrt{1 + x} -\sqrt{ 1 - x}$

I know I have to find the derivatives of $ f(x) $ (i.e. $f'(x)$ ..) but I'm confused about what to do afterwards .
-1
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2answers
173 views

Given $f(x+1)=x^2-3x+2$, how can I find $f(x)$? [closed]

Given $f(x+1)=x^2-3x+2$, how can I find $f(x)$?
0
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1answer
49 views

Bijection question

I need to show that $(m,n) \mapsto 2^{m-1}(2n-1)$ is a bijection of $\mathbb{N} \times \mathbb{N}$ on $\mathbb{N}$ I think I need to show that the expression is both injective and surjective, but I ...
2
votes
1answer
285 views

The domain of fractional exponents

Take the following: $$f(x) = x^{6/4}$$ The domain of this function is all real numbers. This function can be simplified to: $$f(x) = x^{3/2}$$ The domain of this function is all real numbers ...
0
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1answer
39 views

What function space do $u(x)=\sin(x)$ and v$(x)=e^{-2x}$ belong to?

I'm trying to follow notes, a question is presented in the middle of the notes that isn't answered. $u(x)=0$ when $-1\leq x<0$ $u(x)=x^2$ when $0\leq x \leq 1$ The $u(x)$ function belongs to ...
0
votes
1answer
2k views

Given f(x) and g(x), find fg(x) and its domain and range

We are given: $\eqalign{ & f(x) = x - 1 \cr & g(x) = {x^2} \cr} $ Given these functions the answer I get is: $fg(x)=x^2-1$ As the range of $g(x)=x^2$ is always positive this ...
0
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1answer
82 views

What is the Mirror/PingPong clamp mode algorithm?

I do programming as a hobby, and in a dynamic system various numerical values inevitably change. Those values can be greater than or less than the expected range, in which case they need to be ...
0
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2answers
90 views

Number of possible value for $\tan \{x\} = 1$.

The number of solution of equation $\sin \{x\} = \cos \{x\}$ in the interval $[-2\pi,2\pi]$ is, where $\{x\}$ denotes the fractional part of $x$. My thoughts: Dividing both sides by $\cos\{x\}$ we ...
0
votes
1answer
95 views

What are the methods to find inverse of a function?

What are the different methods to find inverse of a function? The one that I have been taught in high school is by converting $y=f(x)$ into $x=y(x)$. But, this requires a lot of simplification and may ...
0
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1answer
19 views

Geometric interpretation of a critical point, i.e. of $q(t) := f(x + t(y-x))$.

So, I know what critical points are. But hear me out on the following notes I made: For $x,y\in \mathbb{R}^n$ we define $$q(t) := f(x + t(y-x)), $$ then $$q'(t)=\nabla f(x+t(y-x))^T(y-x).$$ Now, if ...
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2answers
1k views

How can a unit step function be differentiable??

Recently, I am taking a Signal & System course at my college. In all of the signal & system textbooks I have read, we see that it is written " When we differentiate a Unit Step Function, we ...
1
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1answer
57 views

Whats the name of this function?

I read this function in an exercise. It looks quit familiar to me, however I do not know its name. Whats the name of the $\rho_n$ function and who brought it up first?
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1answer
37 views

Why isn't a given relation a function from A to A?

I'm having some difficulties solving the question below Let $A = \{1, 2, 3, 4\}$ and $C = \{P, Q, R \}$. Why isn't $\{\langle P, P \rangle, \langle Q, R \rangle, \langle R, Q \rangle, ...
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1answer
29 views

How do you find out what the function $g(f(2))$ and $f(g(2))$ is?

I'm trying to find what the *g*$(f(2))$ and the f $(g(2))$ is. Here are the functions for f and g: Let A - $\{$1, 2, 3, 4$\}$ and B - $\{$a, b, c, d$\}$ Let f : A $\rightarrow$ B be defined so ...
1
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1answer
49 views

How can I find the domain and the range of a function?

I'm working on a task which has the following question: What is the domain and the range to the function g? Here is the function g: Let A - $\{$1, 2, 3, 4$\}$, B - $\{$a, b, c, d$\}$ ...
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votes
1answer
259 views

Nondecreasing function

$a)$ Let $f(x)=x−1−\ln x$. Show that $f$ is nondecreasing on $[1,\infty)$. $b)$ Use the result from $(a)$ to show that $$\ln x ≤ x-1\text{ when }x ≥ 1.$$ $c)$ Use the result from $(b)$ to show that ...
0
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1answer
58 views

A nice functional equation

Let $f: \mathbb{R^2} \rightarrow \mathbb{R}\setminus\{0\}$ a function such that $$f(x, y)f(y, z)=f(x, z).$$ Show that $\displaystyle f(x, y)=\frac{h(x)}{h(y)}$.
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2answers
56 views

What restrictions are there on explicit equations?

So I've always been told that for a function to be considered explicit it can only have one specific output for each input or simply pass the vertical line test. While I can accept that on it's face I ...
1
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0answers
43 views

Constructing continuous Functions based on Unbounded Sets.

The question asks the following: "For each of the sets, construct a continuous function that is unbounded on the set.", However, I am completely lost on what I should do or even begin. Some of the ...
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1answer
85 views

Which of the following are correct definitions of functions?

I don't understand this question at all. For (a) and (b), the two equations are on separate lines in a curly bracket - I wasn't sure how to format this so I just separated them using a semi-colon ...
1
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0answers
63 views

Time constructible, non-time constructible functions

A function T:N→N is time constructible if T(n)≥n and there is a Turing Machine M that computes the function x↦└T(|x|)┘ in time T(n). (└T(|x|)┘ denotes the binary representation of the number ...
2
votes
2answers
315 views

Prove that limit goes to infinity if a convex function's derivative > 0

I do understand what a convex function is and I can see geometrically that following statement is true: $(1)$ If for a given convex function $f$ which is differentiable over $\mathbb{R}$, if ...
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0answers
75 views

How find this function $f_{n}(x)$

let $f(x)=2x^2-1,f_{2}(x)=f(f(x)),f_{3}(x)=f(f_{2}(x))=f(f(f(x))),\cdots,f_{n}(x)=f(f_{n-1}(x))$ Find the $f_{n}(x)=$ This problem is from china QQ,and I try I think use $$\cos{2x}=2\cos^2{x}-1$$ ...
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2answers
565 views

How to find out if a function is surjective or injective?

I'm working on task which i'm a bit stuck at. I need to find out which of these functions below are injective and surjective. Let A - $\{$1, 2, 3, 4$\}$, B - $\{$a, b, c, d$\}$ and C - $\{$P, Q, ...
5
votes
1answer
133 views

Maps with every point being periodic

Does there exists a characterization of continuous maps $f:[0,1]\rightarrow [0,1]$ with every point $x\in [0,1]$ being periodic (i.e. if for every $x\in [0,1]$ there exists $n\in\mathbb{N}$ such that ...
0
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2answers
326 views

What's the importance of continuous functions and continuity?

While studying calculus, I've read about continuous functions but I still couldn't figure out what's the importance of the concept, I imagine that the concept (and also the concept of continuity) may ...
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1answer
47 views

Is the statement $a+b < c+ d \implies e^{-a} + e^{-b} > e^{-c} + e^{-d}$ true?

As the title says, I am trying to ascertain whether the following is true: Suppose $a,b,c,d\in \mathbb{R}^+$ are such that $a + b < c + d$, then it is also true that $e^{-a} + e^{-b} > e^{-c} + ...
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3answers
47 views

Help Needed on derivates and functions including maximum and minimum points

How do I solve this question? The function $$f(x)=−4x^3−12x^2+96x+1$$ is increasing on the interval(s) is decreasing on the interval (s) The function has a local maximum at x is How can I solve ...
4
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2answers
356 views

Is this a function and injective/surjective question

Consider the set $A={(x^2, x):x \in R}$. Is this a function from $R$ to $R$? I know it will be a function if there is a unique output per input, but I've never seen a function formatted like this. ...
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1answer
91 views

Understanding a question on iterated logarithms

I have in front of me a math problem that I do not understand. That's to say, I don't understand what is being asked of me. Problem: We can define $\log_2**(x) = log_2*(log_2*(x))$ and the function ...
0
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1answer
64 views

Does $f(\mathbf x _1 + \mathbf c ,…,\mathbf x _n + \mathbf c)=f(\mathbf x _1 ,…,\mathbf x _n)$ imply…

I'm trying to prove the following claim: Let $\mathbf x _1,...,\mathbf x_n\in \mathbf R ^p$ and $f:\mathbf R ^p \times ... \times \mathbf R ^p \ \ \text{(n times!)}\rightarrow \mathbf R.$ Suppose ...
1
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2answers
145 views

How can I optimise the power series calculation of the exponential function?

In an answer to the question Fastest way to calculate $e^x$ upto arbitrary number of decimals? there is a description of a method by which the number of terms needed to calcluate $e^x$ to a given ...
2
votes
2answers
49 views

Function by properties

I am looking for a function $f(x)$, which is vertical at the origin ($f'(0)\approx -\infty$) goes to zero further from the origin ($f(x) \to 0$ for $x\to \infty$ and for $x\to -\infty$) and is ...
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1answer
139 views

Is this function periodic? [closed]

Is the following function periodic? $$f(x)=\cos(x)*\cos(x\sqrt5)$$ A function $f$ is said to be periodic with period $P$ ($P$ being a nonzero constant) if we have $$f(x+P) = f(x)$$ for all ...
4
votes
3answers
369 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 ...
0
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2answers
573 views

How do I prove that a function grows faster than another? [closed]

I need to prove that one function, say $n$ grows faster than say, $\sqrt{n}$?
2
votes
0answers
46 views

problem with submersion

Given $\varphi:\mathbb{R}^{m+n}\longrightarrow \mathbb{R}^m$ is $C^{ k}$ class. If there $a\in \mathbb{R}^{m+n}$ with $\varphi^{\prime}(a)$ is surjective. Then there a mergullo $f:V\longrightarrow ...
1
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0answers
43 views

measure and integral chapter 9

(a)Show that the function h defined by $h(x)=e^{-1/x^2}$ for x>0 and h(x)=0 for $x \le 0$ is in $C^\infty $ (b) Show that the function $g(x)=h(x-a)h(b-x), a<b, is C^\infty$ with support [a,b]. ...
0
votes
1answer
212 views

Does log $f(n) = O($log $g(n))$ imply $f(n) = O(g(n))?$

Assuming log is base 2, if I know that: log $f(n) = O($log $g(n))$. Does this imply that $f(n) = O(g(n))$? I understand that the converse is true.
1
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2answers
81 views

Why are $R^n$ treated as $R^{n+1}$ spaces in $R^{n+1}$?

$y = x$ is a line in $R^2$ space. But if you graph $z = x$ in $R^3$ space, it's a plane: Both functions have the same relations, so why is one a plane but the other a line?