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

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Definition of Lebesgue integrable function

If a function $f : \mathbb{R}^d \to [-\infty,\infty]$ is Lebesgue integrable, then by definition we have $$\int_{\mathbb{R}^d} |f(x)| \, dx < +\infty.$$ Is it possible to say that there exists a ...
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2answers
34 views

Solving Trigonometric Identities - Thinking questions

The question I have is a thinking question: If $\sin(x+y)=0.9$ and $\sin(x-y)=0.6$, determine $\sin x \cos y$. I am really not sure how to go about it. Could I use the addition formula of sin and the ...
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1answer
26 views

Series of random numbers on a continuous function

At one point, I read about a function used to generate random numbers that follow a continuous pattern. By this I mean random numbers that as a series is random, but in which items tend to be ...
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0answers
32 views

Prove that if $\lim_{x\to c}f(x)=L$, then $\lim_{x\to c}|f(x)|=|L|$ [duplicate]

Let I $\subseteq \Bbb R$ be an open interval, let c$\in$I, and let f:I-{c}$\rightarrow\Bbb R$ be a function. Let L $\in\Bbb R$. Prove that if $\lim_{x\to c}f(x)=L$, then $\lim_{x\to c}|f(x)|=|L|$
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13 views

Let $J⊆I⊆\Bbb R$ be open intervals, let $c∈J$, and let $f:I\setminus\{c\}→\Bbb R$ be a function.

Let J⊆I⊆$\Bbb R$ be open intervals, let $c∈J$, and let $f:I\setminus\{c\}→\Bbb R$ be a function. Prove that $\lim_{x \to c}⁡f(x)$ exists if and only if $\lim_{x→c}⁡f/_{J}(x)$ exists, and if these ...
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2answers
37 views

Function, Relation, Operation and Cartesian Product

An operation is a kind of function. A function is a kind of relation. A relation is a subset of a Cartesian product. A Cartesian product is an operation. Back to 1. It seems to me that there's ...
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4answers
69 views

$f :\mathbb N \to \mathbb R$ be the function $f(0)=0 , f(n)=\dfrac 1 n , \forall n >0$;is $\mathbb N$ induced with the metric $|f(x)-f(y)|$ compact?

Let $\mathbb N$ be the set of non-negative integers and $f :\mathbb N \to \mathbb R$ be the function $f(0)=0 , f(n)=\dfrac 1 n , \forall n >0$ , then obviously $f$ is injective , so $d : \mathbb N ...
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3answers
93 views

what is the period of cos (sin nx)? [closed]

Could anyone tell me what the period of $$f(x) = \cos ( \sin {nx}); \qquad n \in \mathbb{N} $$ is.
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1answer
34 views

Constructing a quasiconvex function

Let $C\subset\mathbb{R}^2$ be a nonempty convex set. A function $f:C\rightarrow\mathbb{R}$ is called convex if $$ f(\lambda u+(1-\lambda)v)\leq\lambda f(u)+(1-\lambda)f(v), \quad\forall u,v\in C, ...
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4answers
49 views

Proving trigonometric identity $1+\cot x\tan y=\frac{\sin(x+y)}{\sin x\cos y}$

$$1+\cot x\tan y=\frac{\sin(x+y)}{\sin x\cos y}$$ I have worked through most of this question, and I believe I am so close to finding the answer, but I have run into some issues where I am not sure ...
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2answers
34 views

limits involving a piecewise function, Prove that if $c \ne 2$, then f does not have a limit at $x = c$.

$$f(x) = \begin{cases} (x-2)^3 & \text{if $x$ is rational } \\ (2-x) & \text{if $x$ is irrational } \end{cases}$$ (i) Prove that if $c \ne 2$, then f does not have a limit at $x = c$. (ii) ...
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3answers
31 views

Prove the trigonometric Identity involving secant

The question I am currently working on is: $\sec^2x-2\sec x\ \cos x+\cos^2x=\tan^2x-\sin^2x$. Okay, judging by the expression here, I am going to need to work with the left side of the equation ...
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2answers
19 views

$\sin x \sin(\pi/2-x)=\sin x \cos x $?

During a question involving proving a trigonometric identity, I was given help in which one of the lines showed that $\sin x \sin(\pi/2-x)$ equals $\sin x \cos x$? Could anyone please explain to ...
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2answers
44 views

Proving Trig. Identities [duplicate]

$\cot x=\sin x \sin(\pi/2 -x) + \cos^2x \cot x$ I'm having difficulty with figuring out how to prove trigonometric identities. I know that in order to do these you need to use the trig ratios ...
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1answer
38 views

Proving trigonometric identities

Prove: $\cot x=\sin x\sin(\pi/2-x)+\cos^2x\cot x$. Hi there! So this problem asks to prove this trigonometric identity. I am not sure how to approach these problems other than needing to know the ...
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2answers
24 views

Simplify the expression using trigonometric identities

Simplify: $$\frac{\sin(x)\cos(x)}{1-\sin^2(x)}$$ This looks to be similar to the Pythagorean identities: $\cos^2(x)=1-\sin^2(x)$. However, I am not certain about how to approach this. I'm thinking ...
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1answer
27 views

Let $f(m,1) = f(1,n) = 1$ for $m \geq 1, n \geq 1,$ and let $f(m,n) = f(m-1,n) + f(m,n-1) +

Let $f(m,1) = f(1,n) = 1$ for $m \geq 1, n \geq 1,$ and let $f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1)$ for $m > 1$ and $n > 1.$ Also, let $$S(n) = \sum_{a+b=n} f(a,b), \text{ for } a \geq 1, b ...
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1answer
43 views

Proving set of bounded continuous functions is an open set

appreciate your help with the below: Question: Let C[0,1] be the set of continuous functions from [0,1] to $\mathbb{R}$. Consider the metric space M = (C[0,1],d) where d denotes the sup metric. ...
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1answer
95 views

$\max(a,b)=\frac{a+b+|a-b|}{2}$ generalization

I am aware of an occasionally handy identity: $$\max(a,b)=\frac{a+b+|a-b|}{2}$$ However, I have found I'm unable to come up with a nice similar form for $\max(a,b,c)$. Of course I could always use ...
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2answers
72 views

Find the exact value of $\cos(11\pi/12)$.

This question I look at as being similar to $\sin(7\pi/12)$. You can break it up using the special triangles into $3\pi/12 + 4\pi/12$. However with this one, I can't find one of the angles in which ...
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2answers
23 views

Finding Exact values using compound angle formulae

Find the exact value of each expression: 1) $\sin{(-\frac{\pi}{2} +\frac{\pi}{3})}$ -For this question, it would appear as though you could use the addition compound angle formula ...
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1answer
25 views

Which of these relations are maps?

List all relations $\{a,b\} \to \{c,d\}$, assuming $a \neq b$ and $c \neq d$. Which of them are maps? So I know the cartesian product gives $\{(a,c),(a,d),(b,c),(b,d)\}$. And the relations will be ...
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0answers
25 views

A question regarding rational functions being onto.

We are given the following function: $$K:\Bbb{R}\setminus \{2\}\mapsto \Bbb{R}\setminus \{1\}$$ defined by $K(x)=\dfrac{x+12}{x-2}$ Is the function K onto (surjective) or not?
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1answer
49 views

Is monotonic function implies its invertibility and vice-versa?

A monotone function never has saddle points, same is true for an invertible function. Can we conclude that monotonic function is also invertible and vice-versa?
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1answer
10 views

Simply function F and find alpha for which F will be min

I have point coordinates like [x, y], where x and y are positive natural numbers. I need to ...
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1answer
19 views

On the function $f\colon(-\pi/2,\pi/2)\rightarrow (-\infty,\infty)$ given by $x\mapsto 1+2x$

If $x\mapsto 1+2x$ is a function having $(-\pi/2,\pi/2)$ as domain and $(-\infty,\infty)$ as codomain then it is onto but not one one one one but not onto one one and onto neither one ...
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1answer
94 views

Let $f\colon [0,1]\to\mathbb{R}$. Prove there exists $c$ such that $f(c)=c$

Let $f\colon [0,1]\to\mathbb{R}, 0<f(x)<1$ for all $x \in[0,1]$. $f(x)$ continuous. Prove there exists $c$ such that $f(c)=c$ My attempts: As $x \in[0,1]$ then $0\leq x \leq 1$ so $0 < x ...
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2answers
44 views

Exact values of $x$ for $2\cos^2x=1+\sin x$ [duplicate]

This question involves finding the exact values of $x$ such that $0 \leqslant x \leqslant 2\pi$. So far I have subtracted everything to the left side of the equation and then used the pythagorean ...
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3answers
31 views

Determine the exact value of equations involving more two trig variables

$2\cos^2x=1+\sin x$. Determine the exact values of $x$ such that $0 \leq x \leq 2\pi$. I am experiencing problems with factoring this question. First I started by getting everything on to the ...
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2answers
21 views

Prove that $f$ has an inflection point at zero if $f$ is a function that satisfies a given set of hypotheses

Prove that if $f$ is a function such that $f'(x) > 0$ $\forall x \neq 0 \wedge f'(0) = 0 \wedge f''$ is a continuous one to one function on some open interval $(a, b): a < 0 < b$ then $f$ ...
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0answers
18 views

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$ I need to find the following: $(a)$ Show $F$ is one-to-one on $A$. $(b)$ Show that $F(A) = \{(u,v) : 0 < \frac{u}{2} < v ...
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0answers
47 views

How can we show that the functions are differentiable?

Show that the following functions $$f(x, y)=\frac{xy}{\sqrt{x^2+y^2}} \\ f(x, y)=\frac{x^2y}{x^4+y^2}$$ are differentiable at each point of the domain. Determine which of them is $C^1$. $$$$ The ...
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1answer
36 views

Determining the exact value of trigonometric functions using tan

Use the special triangles to give exact solutions where possible. Find all values of x such that $0 \le x \le 2\pi$. The question I have is $\tan^2x=1$. What I have done so far (it appears that ...
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1answer
24 views

Difficulty in understanding a piecewise function problem

I am using Stitz-Zeager PreCalculus book and I am not able to fully understand the problem. The Part I am feeling problem in is the c one. The Problem is as follows: original image For $n$ copies ...
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1answer
22 views

Special Triangles and their related acute angles

So I've been working on some questions involving having to find the exact value of trig. functions involving a particular interval. I have worked through the question but now I have something I am ...
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1answer
27 views

Find all values of $x$, linear and quadratic functions

Use special triangles to give exact solutions where possible. Find all values of $x$ such that $0 \leq x \leq 2 \pi$. 1) $\cos^2 x + \cos x - 1 = 0$ For this question, I have factored in which the ...
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0answers
32 views

How is $\cos^3{x}$ an odd function while $\sin^3{x}$ an even function?

We know that for odd function $f(-x) = -f(x)$ and for even function $f(-x) = f(x)$. Therefore, $\cos^3(-x) = \cos(-x)\cos(-x)\cos(-x) = \cos{x}\cos{x}\cos{x} = \cos^3{x}$ (i.e. $\cos^3{x}$ must be ...
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2answers
25 views

Inverse Image Proof

Let $f:X\rightarrow Y$. Let $A$, $A_1$ and $A_2$ be subsets of $X$ and $B$, $B_1$, and $B_2$ be subsets of $Y$. Then, I need to prove that $f^{-1}(B_1\cup B_2)=f^{-1}(B_1)\cup f^{-1}(B_2)$. I know ...
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1answer
47 views

Show that $f(a,b)$ is one-to-one

Let $$A=\{(x,y)\in\mathbb R^2:x>0, y>0\}$$ and define $f:A\to\mathbb R^2$ by $$f(a,b)=(a+b^2,2a^2+b).$$ Show that $f$ is one-to-one on $A$. I know that a function is one-to-one if all ...
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3answers
47 views

Prove $\lim_{x\to\infty} \left( \sqrt{x+1} - \sqrt{x} \right) = 0$

My attempt: I tried manipulating the formula, but I couldn't do anything useful. I tried to find another function $f(x)$ such that $\lim_{x\to\infty} f(x) = 0$ and $f(x) \geq \sqrt{x+1} - \sqrt{x} $ ...
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1answer
20 views

How to Prove this Fractional Linear Transformation of $\mathbb C$ takes $S^1$ to itself?

Let $x\in\mathbb C$. I know that $|x|<1$ but I don't think that matters for what I'm about to ask. Let $f$ be the fractional linear transformation $f(z)=\frac{z-x}{1-\overline x z}$. Then I'm ...
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0answers
29 views

Proof that set of functions with derivative zero at a given point is meager in space of strictly increasing twice differentiable functions.

Let $X = \{f: [0,1] \to \mathbb{R} \; | \; f\in C^2[0,1], f \textrm{ strictly increasing} \}$. I equip $X$ with the topology of uniform convergence. Define the set $A$ as: $$A =\{ f \in X \; | \; ...
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0answers
11 views

“Sigmoid” function with tunable initial slope, upper asymptote and transition period

I'm looking for a function which resembles the transition between the function $f(x)=x$ for small $x$ and the function $f(x)=C$ for large $x$ ($x$ is finite and $\geq 0$). I've found the generalized ...
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2answers
32 views

How to find unknown variables?

Given that $f(x)= 8x+m$ and $g(x)= x^2 - 3x +n$ and $g\circ f(x) =64x^2-8x$ where $m $ and $n$ are constants. Find the values of $m$ and $n$. Can someone point me in the correct direction on how ...
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5answers
93 views

Range of $\cos(2 \sin x)$ [closed]

Actually, I am confused determining the range of the function given below. Could anyone tell me what the range of $ f(x) = \cos (2 \sin x)$ is ?
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0answers
22 views

Integral of a function depending on two variables

I have a function that depends on two parameters, say $f(x,y)$. Now I need to integrate this function from $0$ to $y_{max}$. I have all the values of $x$ and $y$ and also the values of this function ...
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4answers
48 views

Are all functions that have an inverse bijective functions?

To have an inverse, a function must be injective i.e one-one. Now, I believe the function must be surjective i.e. onto, to have an inverse, since if it is not surjective, the function's inverse's ...
1
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1answer
27 views

What is the name of the most locally convex / concave point of a $f(x)$ function?

I was looking for the name given to the more locally convex / concave points of a given function $f(x)$ for instance, the ones I have marked in the multiplicative inverse function below. In the case ...
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3answers
255 views

Transforming a function by a sequence geometric operations on its graph.

I am solving the following problem: Let $f(x) =\sqrt{x}$. Find a formula for a function $g$ whose graph is obtained from $f$ from the given sequence of transformations: shift right $3$ ...
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0answers
22 views

Expectation of function and its derivatives

I was wondering if anyone has an idea of what could be said about $f$ given that it satisfies the following inequality: $$\mathbb{E}\left[ \frac{1}{\sigma^2}f(x)^2+f'(x)^2+2f(x)f''(x) \right] \geq 0$$ ...