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

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1answer
19 views

Differentiation and 3 dimensional

Assume that a Mountain is shaped like a function $z=f(x,y)=sin(xy)$. A Hiker begins at the Point $(0,0,0)$ and wants to reach the Point (1,1,sin(1)), but he is not allowed to get over the Slope of ...
0
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1answer
49 views

Functional Equation $\frac {f(x)}{f(y)} \le 2^{(x-y)^2} $

The function $f$ satisfies $$ \frac{f(x)}{f(y)} \le 2^{(x-y)^2}$$ for all $x,y$ in the domain of $f$. Then $f$ can be which of the following: (A) $\sqrt{x}+x^3$ (B) $\int_0^{\sin^2 x} ...
0
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0answers
19 views

Expanding a function around $x=\infty$ (Puiseux series)

I am trying to expand the function $$f(x) = x(2x^2-3)(x^2+1)^{1/2}+3sinh^{-1}(x)$$ around x equals infinity but don't know where to begin. I have found some articles online that talk about using a ...
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0answers
11 views

Clarification on a reflexive function

$R_1 = \{(a, b) | a ≤ b\}$ is a reflexive function, but I'm confused on why it is. $a≤b$ but doesn't that not necessarily mean that there is an $a$ that will equal $b$? Couldn't all of the $a$ very ...
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3answers
38 views

Is the inverse of a real, continuous “1-1” function necessarily continuous itself? [closed]

If so, please do provide me with an epsilon-delta proof, if possible. Thanks in advance.
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0answers
15 views

Find the complex potential given the real part $\frac{6000}{\pi}\theta$

In my 'Complex Variable Theory' notes, we have an example where we use a linear fractional mapping to solve for Laplace's equation between semi-circular plates. The Question reads: Find the ...
0
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2answers
27 views

How to calculate $\int_{S^1}\frac{1}{|w-z|^2}ds(w)$

$S^1$ is the unit circle on the complex plane,$ds$ is the normalized lebesgue measure on $S^1$, $z\in \mathbb{D}$ which is the open unit disk on the complex plane. How to calculate ...
2
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1answer
61 views

How to calculate $\lim\limits_{x\to 0}\frac{\int^{2x}_{x}g(t)dt}{x^2}$ assuming $g(0)=0$?

Assume $g(x)$ is differentiable everywhere .So $\lim\limits_{x\to 0}\frac{\int^{2x}_{x}g(t)dt}{x^2}=\lim\limits_{x\to 0}\frac{\int^{2x}_{0}g(t)dt-\int^{x}_{0}g(t)dt}{x^2}$. But the problem here is how ...
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0answers
13 views

Local form of the function

I am trying to solve this problem: One of the contours (i.e. loci of locations with the same value) of a generic smooth scalar function of the two-dimensional plane is roughly figure-of-eight in ...
2
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3answers
41 views

About the nature of continuity of trigonometric functions and equality

I was recently somewhat confused by the result of an exercise from a textbook that read: Question How many solutions are there to the equation $(\tan x)\sin^2(2x)=\cos x$, $-2\pi \leq x \leq ...
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1answer
13 views

Relations and restriction of a function.

This is a homework question: "Let R be an equivalence relation on a set S. For A ⊆ S, we define RA to be the restriction of R to elements of the set A, i.e., RA is a relation on A such that for any ...
1
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0answers
32 views

How to see a function is $C^2$

Let $\phi_{z}(w)=\frac{z-w}{1-w\bar{z}}$ which is a conformal mapping of $\mathbb{D}$. $f\in C(\bar{\mathbb{D}})\cap C^2(\mathbb{D})$. $\mathbb{D}$ is the unit disk centered at the origin in the ...
0
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1answer
28 views

Determine if $f=\{(x,y)\mid 2x+3y=7\}$ is invertible. From $\mathbb R \rightarrow \mathbb R$. If it is invert it.

I am thinking this is no, because I am not even sure if this counts as a function? I am unsure how this can be a function if there exist only a few $(x,y)$s that fulfill the equation. Or does the ...
0
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0answers
27 views

“Inverse” of a step function

How can I write the function below $$f(x)=\left\{\begin{array}{ll} 1, & 0\leq t\leq 1, \\ 0, & t>1 \end{array}\right.$$ using the unit step function? I mean, I don't know how could I write ...
0
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0answers
43 views

Proving h(x) = f(x)+g(x) is one-to-one where f and g are one-to-one functions

I have attempted to solve the question below, but I am not sure if it is correct. Let S = {1,2,3,4} and let F be the set of all functions from S to S. Let R be the relation on F defined by: For all ...
-4
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1answer
29 views

Need hint on differentiable exercise [closed]

I'm looking for a sequence of continuously differentiable $f_n : [0,1] \to \mathbb R$ functions, such that $f_n \rightarrow 0$ uniformly, but $f'_n$ doesn't converge uniformly. Could someone give me ...
0
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2answers
15 views

How to prove (given $f\colon A\to B$ and $g\colon B \to C$), if $g∘ f\colon A \to C$ is onto, then $g$ is onto; if $g∘ f $ is 1to1, then $f$ is 1to1

I am doing practice problems for my exam, and I can't seem to figure this one out. Let 𝑓: 𝐴 → 𝐵, 𝑔:𝐵 → 𝐶. Prove that: (a) if 𝑔 ∘ 𝑓: 𝐴 → 𝐶 is onto, then 𝑔 is onto (b) if 𝑔 ∘ 𝑓: 𝐴 → 𝐶 ...
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2answers
33 views

Proving that $f$ is the identity function if $ \ f \circ g = g \circ f$ for all $g$.

This is a problem from Spivak's Calculus. I can see how it applies for some family of functions, for example, if $g$ is constant; but I need to prove it for any $g$.
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1answer
19 views

Correct notation with composite function & characteristic functions.

I have the functions $p: \mathbb R \to \mathbb R; \quad p(x) = \frac12 x + 1$ $q: \mathbb Z \to \{0, 1\}; \quad q(x) = \begin{cases} 1 & x \geq 1 \\ 0 & x \leq 0 \end{cases} $ I know that ...
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4answers
52 views

Dependence and Independence of $\epsilon$ and $\delta$

This is a question regarding Epsilon-Delta proofs, in this example in Single-Variable Calculus, but hopefully the crux of what I'm asking here, will be general enough. Introduction I'm putting ...
1
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1answer
30 views

In the function $r :\mathbb{R} − \{0\} \to \mathbb{R}$, what effect does the “$-\{0\}$” have on the domain?

The full function is: $r : \mathbb{R} − \{0\} \to \mathbb{R}$ defined by $r(x) = 6 / x$.
0
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1answer
21 views

Inverse image of function

Let $f:A\to B$ be a function such that $A= \{1,2,3\}, B=\{1,2,3\}$ $f(1)=1 , f(2)=1, f(3)=2$ Then is the inverse image of $f^{-1}(B) = \{1,2,3\}$? And, $f^{-1}(1) = \{1, 2\}$ but is it okay ...
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2answers
21 views

Can a function have a set as argument or value?

I was joking with someone when I spelled some proper noun in a mathematical equation: $$\tan x \in f(u)$$ Obviously this function needs to have a value that is a set in order to make sense. Hence, ...
1
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1answer
31 views

how to find mininimum $f(x)$ using $\int_{-\infty}^{\infty} f(x)g(x)dx$?

I would like to know the $f(x)$ which minimizes the $\displaystyle\int_{-\infty}^{\infty} f(x)g(x)\,dx$. Actually, this question start from the MMSE (Minimize Mean square error) ...
0
votes
1answer
16 views

Proving the discontinuity of $g(x)=\sqrt{2+\tan^2x}$ without the aid of a graph

How would you show where the discontinuity of $g(x)=\sqrt{2+\tan^2x}$ is without a graph? What sort of approach would you take mathematically? For example, using the method for piece-wise functions. ...
0
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2answers
35 views

Let $f(x)$ be an increasing function. Assume its image $f(C) $ is also a connected set. Prove that $f$ must be continuous

Let $f: R → R$ be an increasing function. Assume that for every connected subset $A$ of $R$, its image $f(A)$ is also a connected set. Prove that $f$ must be continuous. To prove this, I am thinking ...
1
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2answers
46 views

Prove if a continuous function $f$ is one-to-one, it is monotonic.

This is the converse of Prove that if function f is monotonic, then it one-to-one. Let $f:(a,b) \mapsto \mathbb{R}$ be a continuous function. Prove if $f$ is continuous in $(a,b)$, $f$ is monotonic. ...
2
votes
4answers
123 views

What is $\lim\limits_{n\to {\infty}} (\frac{n}{1+n})^n$? [duplicate]

What is $\lim\limits_{n\to {\infty}} (\frac{n}{1+n})^n$. Is it possible to write the function $f(x)=x^n$ and since we know $\frac{n}{1+n}\to 1$, so $f(\frac{n}{1+n})\to 1^n=1$. So the limit it $1$. ...
0
votes
1answer
37 views

Having trouble understanding how to disprove/prove if a formula is a function.

Is $\frac 1{x^2-2} $ a function from $\mathbb{R}\to \mathbb{R}$? Is it a function from $\mathbb{Z}\to \mathbb{R}$? I have been thinking about this but, I can't find any example for which you can have ...
0
votes
0answers
10 views

Time taken to run a function R(n,a)

R(n, a){ if n = 1 return(a); if n > 1 return (R(n − 1) + R(n − 1) + 1); } Could you please explain me why the estimated time taken to run R(n, a) as a function of n is: (2^(n−1))*(a + 1) − 1 ? ...
0
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3answers
33 views

Is this true :if $x\in [0.2] $ then $f(x)=\frac{2x+3}{x+2} \in [0.2]$?

I'm sorry to ask this question mayeb it's a trivial question but i would like to confirme if i have this function $f(x)=\frac{2x+3}{x+2}$ which $x$ is a real number in $[0.2] $ then $f(x) \in [0.2]$ ...
1
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0answers
36 views

Prove there is a point $y$ such that $ g(y)=0 $ where $ g: \bar{D}^2 \rightarrow \mathbb{C} $

Let $ g: \bar{D}^2 \rightarrow \mathbb{C} $ be a continuous function on a closed disk satisfying $ g(-x)=-g(x) $ for any $ x \in \partial \bar{D}^2$ Prove that there is a point $ y \in ...
1
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1answer
40 views

Examples of complex-variable functions that fail to have a limit at some point

My notes from class have the example $\frac{\overline{z}}{z}$ as z tends to zero. That the limit does not exist is shown by exhibiting that along the $x$-axis the limit is $1$ and along the $y$-axis ...
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0answers
16 views

Suppose f(z)= u(x,y) + iv(x,y) is a nonconstant analytic function

please I need someone to help me to solve this problem. Indeed I could not understand what the question wants? I am confused thanks. Suppose f(z)= u(x,y) + iv(x,y) is a nonconstant analytic function ...
5
votes
4answers
300 views

Different functions or same functions

I have a question in my booklet : $f(x) = \frac{x}{x}$ and $f(x) = 1$ are different or not why or why not? I can only think that the functions are different because the second one is a constant ...
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0answers
31 views

Find the number of points of distance n away from origin as function of n

I came across a seemingly simple problem the other day and I thought I'd share it with anyone interested. Say you have a point in 3 dimensions. The number of points that are of distance $0$ away is ...
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1answer
34 views

twice differentiable functions

Let $f:\mathbb R \to \mathbb R$ be a twice continuously differentiable function, with $f(0)=f(1)=f'(0) = 0$. Then $f^{"}$ is the zero function. $f^{"}(0)$ is zero. $f^{"}(x)=0$ for some $x \in $ ...
0
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1answer
54 views

A question that I could not solve.

Let $x \in (0,1) \setminus \mathbb{Q}$ and $y \in (1,\infty) \setminus \mathbb{Q}$ , $xy \neq 1$. We have ${\bf a}\in {\mathbb{N^+}}^2$, ${\bf a}=(a_1,a_2)$, $f({\bf a})=x^{a_1} y^{a_2}$. What is ...
1
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1answer
35 views

Acceleration: If I know distance, time, and initial velocity, what's acceleration and final velocity?

So I know the Initial Velocity ($V_i$), Time ($t$), and Distance ($d$). I know that $$d = V_it + \frac{1}{2} at^2$$ If I rearrange this, would acceleration $a = \dfrac{2(d - V_it)}{t^2}$ ? Then ...
1
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2answers
40 views

If $p(x)$ is an odd polynomial, prove that $p(x)=x \cdot q(x)$ where $q(x)$ is an even polynomial

Question: A function $f(x)$ is called even if $f(-x)=f(x)$ for all $x$, and odd if $f(-x)=-f(x)$ for all $x$. If $p(x)$ is an odd polynomial, prove that $p(x)=x \cdot q(x)$ where ...
0
votes
1answer
24 views

Let $f:S\to T$. For each $B\subset T$, we have $f[f^{-1}[B]]=B$ iff $B\subset range(f)$

Let $f:S\to T$. For each $B\subset T$, we have $f[f^{-1}[B]]=B$ iff $B\subset range(f)$ I have the following to prove the $\leftarrow$ of the iff: Suppose $B\subset range(f)$. *Then, for some $y\in ...
1
vote
1answer
12 views

Let $f:A\to B$. For each $B\subset T$, we have $f[f^{-1}[B]]=B$ iff $B\subset range(f)$

Let $f:S\to T$. For each $B\subset T$, we have $f[f^{-1}[B]]=B$ iff $B\subset range(f)$ I have the following to prove the $\to$ of the iff: Let $B\subset T$. Suppose $f[f^{-1}[B]]=B$. *Then, ...
0
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0answers
10 views

What can I infer about $h(x,y)$ and $g(z)$ here?

I have the following equation (I'm actually deriving a deformation field from a strain tensor, but that's not important as at this stage it is pure calculus and I'm a little stuck at this stage): $$Ay ...
0
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0answers
12 views

Table of Product Representations of Functions?

Does anyone know where I could find a (preferably free, online) table with infinite product expansions of basic functions (e.g. trig functions, logarithms, special functions)? Specifically ...
0
votes
0answers
20 views

Lipschitz constant and strongly convex parameter of a function?

If $f:\mathbb{R}^n \mapsto \mathbb{R}$ is strongly convex with parameter $m$,i.e. $ f(\boldsymbol{y}) \ge f(\boldsymbol{x}) + \nabla f(\boldsymbol{x})^T(\boldsymbol{y} - \boldsymbol{x}) + \frac{m}{2} ...
0
votes
1answer
24 views

Is a function, in part, defined by it's domain?

I have read the definition of a function from two sources.Both sources state that a function defines a relationship between the input and the output. However, the first source states that it is the ...
-1
votes
1answer
34 views

Calculate how much ethanol to add to petrol to get a desired blend

Take the following problem: Data: I have $20$ liters of petrol in a tank: Assume that e85 is defined as $85\%$ of ethanol and $15\%$ of petrol; Assume that petrol does not contain ...
2
votes
1answer
31 views

If $f$ is an even function defined on the interval $(-5,5)$ then four real values of $x$ satisfying the equation $f(x)=f(\frac{x+1}{x+2})$ are?

If $f$ is an even function defined on the interval $(-5,5)$ then four real values of $x$ satisfying the equation $f(x)=f(\frac{x+1}{x+2})$ are? I thought that $(x+1)/(x+2)=-x$.But I'm getting only ...
1
vote
2answers
33 views

Method of solving the functional equation $f(2x)=f(x)$ using Lagrange's Mean Value Theorem

A problem i have goes as follows: Let $f:\mathbb R\to\mathbb R$ be a continuous function satisfying $f(2x)=f(x),\;\forall\;x\in\mathbb R$. If $f(1)=3$, then the value of $\displaystyle \int_{-1}^1 ...
0
votes
0answers
26 views

If $f(x)=\sin(\log(\frac{\sqrt{4-x^2}}{1-x}))$ then the range of $f(x)$ is? [duplicate]

If $f(x)=\sin(\log(\frac{\sqrt{4-x^2}}{1-x}))$ then the range of $f(x)$ is? I found the domain of the function is $-2<x<1$.But I'm having difficulty in finding the range.