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

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1answer
4 views

Inverse function table

I am required to create a table of values (like the one above) for h-1(x). Because x is ordered, i am just wondering, would the two tables would be identical? I just feel a little insulted that's ...
0
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1answer
17 views

group of linear functions and metabelian groups

Let $G$ represent the group of linear functions under composition of the form $x \mapsto ax+b$ where $a,b \in \mathbb{Q}$ and $a\neq0$. Is $G$ a metabelian group?
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2answers
76 views

To prove that no such function can be continuous.

Suppose $f: [a,b] \to R$ is two to one. that is, for each $y$ in $R$, $f^{-1}({y})$ is empty or contains exactly two points. How to prove that no such function can be continuous.
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2answers
34 views

Show $f(f^{-1}(B_0))\subset B_0$ and that equality holds if $f$ is surjective

Let $f: A\to B$. Let $A_0\subset A$ and $B_0\subset B$. Show $f(f^{-1}(B_0))\subset B_0$ and that equality holds if $f$ is surjective. Attempt: I already did the first part. It is showing that ...
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2answers
24 views

Uniform convergence of sequence of functions with infinite roots to a limit with finite roots

Consider a sequence of continuous functions $(f_n)$ defined over $[0,1]$ such that, for all $n$, the set: $$A_n = \{x\in [0,1] : f_n(x) = 0\}$$ is infinite in cardinality. Can $(f_n)$ uniformly ...
1
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1answer
28 views

Find the range of the function $f(x)$ if $f(x) = 2^x + \frac{4}{2^x}$

I tried this by a logical approach as the sum of two positive numbers is constant will be minimum if they are equal , i.e. $\frac{4}{2^x}$ each should be equal to $2.$ Hence minimum value will be $4.$ ...
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2answers
19 views

Does a one-to-one function exhibit “injectiveness” or “injectivity”?

I'm preparing some tutorials for students and I'm faced with writer's block. If I want to say a function is injective/one-to-one, would the function demonstrate "injectivity" or "injectiveness"? ...
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1answer
25 views

Completely monotonic function intersect

Is there any proof that two "completely monotonic" functions ($f,g: (0, \infty) \rightarrow \mathbb{R}$) would intersect at most at one point? Completely monotonic means: The $n$'th derivative of ...
4
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1answer
57 views

Finding this weird limit involving periodic functions with periods 5 and 10.

If $f(x)$ and $g(x)$ are two periodic functions with periods 5 and 10 respectively, such that: $$\lim_{x\to0}\frac{f(x)}x=\lim_{x\to0}\frac{g(x)}x=k;\quad k>0$$ then for $n\in\mathbb N$, the value ...
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2answers
24 views

Proving a norm on the space of differentiable functions

I consider the space $C^1[a, b]$ of (complex) functions that are at least once differentiable on $[a, b]$. I want to show that $$||f||_{C^1} := ||f||_\infty + ||f'||_\infty$$ defines a norm on ...
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1answer
21 views

Set of all functions between two sets [on hold]

Let $A,B,C,D$ be sets such that $|A| = |B|$ and $|C| = |D|$. Show that $|A^C|=|B^D|$.
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0answers
15 views

Fourier Transform on unit circle? [on hold]

Let $F$ be a continuous function on the close $\overline{\mathbb{D}}$ of the unit disk. Assume that $F$ is in $C^1$ on the (open) disk $\mathbb{D}$, and $\int_{\mathbb{D}} |∇F|^2 < ∞$. Let $f(e^{ ...
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2answers
64 views

Find the value of $x$ that satisfies the equation $\log_{10} \left(\frac{x^{\frac{1}{x}}}{x^{\frac{1}{x+1}}}\right) = 1/5050$ .

I tried it many times and it went bit of lengthy , i reached until \begin{equation*} \log_{10}(x^{1/(x^2+x)}) \end{equation*} then i multiplied $2$ both numerator and denominator and then it is ...
-1
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1answer
29 views

The area of the graph consisting of all the points (x,y) such that $x^2 + y^2 \le 1 \le \left|x\right| + \left|y\right|$? [on hold]

I tried plotting graph of both functions but i am not able to get answer in the form of options given . What is the area of the region in $\mathbb R^2$ such that $$x^2 + y^2 \le 1 \le ...
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0answers
34 views

Find the range of the function : $\frac{1}{\pi}(\sin^{-1}x+\tan^{-1}x) + \frac{x+1}{x^2+2x+5}$ [duplicate]

Problem : Find the range of the function : $\frac{1}{\pi}(\sin^{-1}x+\tan^{-1}x) + \frac{x+1}{x^2+2x+5}$ My approach : Let $g(x) = (\sin^{-1}x+\tan^{-1}x)$ and $h(x)=\frac{x+1}{x^2+2x+5}$ and ...
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0answers
14 views

Composition of differentiable and nowhere differentiable function

This is actually problem 17 from Chapter 10 of the 4th edition of Michael Spivak's "Calculus". The statement is quite simple but I have not had any success in finding an example. Here is the ...
0
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1answer
46 views

Odd or even function?

Is the function $f(x)=-1$ for $-\pi$ to $0$ and $x$ from $0$ to $\pi$ odd or even? How do I determine this for this function? Any help would be much appreciated.
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0answers
23 views

Least squares aproximation

In a Problem of least squares aproximation of a function $f:\mathbb R\longrightarrow\mathbb R$, in an interval $[a, b]$ by a polynomial of degree $n$ ...
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1answer
19 views

Invalid range from inequality

We were given this function and asked to give Range. $$f(x)~=~\dfrac{x^2}{x^2+1}$$ Now I took 3 cases and deduced that $\text{Range} = \left[~0,\infty ~\right)$ Now it is obvious that if we divide ...
0
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1answer
24 views

Inverse of elementary functions

which may be two right inverse of: 1) $h:\Re \rightarrow [0,\infty) $ defined by $h(x)=|x|$ 2) $k:\Re \rightarrow [1,\infty)$ defined by $k(x)= e^{x^2}$
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2answers
34 views

Domain of the given function

A function $y(x)$ is defined as $$ 2^y+2^x=2 $$ The question is about finding it's domain. Pretty simple. By observing the function I could say all the negative numbers are in the domain. But, I think ...
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1answer
20 views

Cardinality of Sets and injections

Let A,B,C,D sets. if |A| $\le$|B| and |B| < |C|, show that |A| < |C| Proof: Case1: suppose |A| < |B| then there exists injection f: A$\to$B and |B| < |C| then there exists injection ...
2
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2answers
58 views

Taking a time derivative of a function of 3 variables.

I have a function of $3$ variables which are all functions of $t$. $$x = \frac{v_1t-y}{\sqrt{(v_2/\dot{x})^2 -1}} \tag 1 $$ In the equation $v_1,v_2$ are constant and $x$ and $y$ are both function ...
2
votes
2answers
57 views

Confusion with seeming lack of notational coherence between $\sin^{-1}(x)$ and $\sin^2(x)$

It seems that $\sin^2(x)$ is used to denote the square of whatever value $\sin(x)$ is, instead of the expected $(\sin(x))^2$. Based on that, I would assume that $\sin^{-1}(x) = \frac{1}{\sin(x)}$, ...
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4answers
61 views

limit of $\ln x + (x+1)/x$ as $x$ approaches $o$

I want to establish monotone intervals of function $f:(0, \infty) \rightarrow \mathbb R$, where $f(x)=(x+1)\ln x$ using its first derivative. I proved that the first derivative of $f$ is an injective ...
-2
votes
1answer
27 views

Assume that p is a real number . In order for $\sqrt[3]{x+3p+1}-\sqrt[3]{x}=1$ to have real solutions, then p [on hold]

Assume that $p$ is a real number. In order for $\sqrt[3]{x+3p+1}-\sqrt[3]{x}=1$ to have real solutions, then $p$: Options A) $p \geq 1/4$ B) $p \geq -1/4$ C) $p\geq1/3$ D) $p \geq-1/3$
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4answers
61 views

Suppose $f$ is a real function satisfying $f(x+f(x))$ = $4f(x)$ and $f(1) = 4$. Then the value of $f(21)$?

Should I proceed with just putting the value of $f(1)=4$ in the first equation or there will be a different way of solving this ?
0
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1answer
19 views

Let $\{a,b,c,d,e,f,g,h\}$ be distinct elements in the set $\{ -7 , -5 , -3 , -2 , 2 , 4 , 6 , 13 \}$ .

Let $\{a,b,c,d,e,f,g,h\}$ be distinct elements in the set$ \{ -7 , -5 , -3 , -2 , 2 , 4 , 6 , 13 \}$ . The minimum possible value of $(a+b+c+d)^2 + (e+f+g+h)^2$ is ? I tried it by making the first ...
1
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1answer
52 views

Can a function be differentiable while having a discontinuous derivative?

Recently I came across functions like $x^2\sin(1/x)$ and $x^3\sin(1/x)$ where the derivatives were discontinuous. Can there exist a function whose derivative is not conitnuous, and yet the function is ...
1
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2answers
35 views

$\sqrt{4x -3}$ injective? Bijective? Inverse?

I'm shown part of the function $g(x) = \sqrt{4x-3}$. Is it injective? I said yes as per definition if $f(x) = f(y)$, then $x =y$. Is this right? Under what criteria is $g(x)$ bijective? For what ...
1
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2answers
20 views

confusion about solving and graphing a simple rational function

given the function: $\frac{x+1}{5} - 2 = -\frac{4}{x}$ I could multiply through by $5x$ yielding the quadratic with solutions $(5,4)$: $x^2 - 9x + 20 = 0$ or.... I could create a common ...
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1answer
14 views

Proof that $f:\mathbb CG \to S$ given by $f(a)=a\cdot s$ is surjective

I am going through a proof of the following result from my lecture notes: Suppose $\mathbb C G \cong \bigoplus_{i=1}^nU_i$ where the $U_i$ are simple $\mathbb CG $ modules. Then any simple $\mathbb ...
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2answers
18 views

Function (How to determine limit by using Maclaurin's series?)

$$ f(x)=\begin{cases} \dfrac{\sin x^2}{1-\cos2x},&-\pi<x<0 \\[1ex] \dfrac{2+\sqrt{x}}{4-\sqrt{x}},&0\leq x<\pi & \end{cases}$$ My question is how to determine whether ...
2
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3answers
38 views

Construct a non-linear function that shows that the intervals $[2,4]$ and $[10,22]$ have the same cardinality

Using something other than a linear function, show the intervals $[2,4]$ and $[10,22]$ have the same cardinality. I don't quite know where to start with this problem, or what key factor is necessary ...
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3answers
122 views

Finding a convex function between two points

Given two points of the $xy$ plane, is there a way to find the equation of a convex function between those two points? I know the answer wont be unique so I'm just looking for a general equation that ...
0
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2answers
42 views

Show that the function $f:\mathbb{Z} \rightarrow \mathbb{2Z}$ by $f(a)=a^3+3a+2$ is not onto

If the function $f:\mathbb{Z} \rightarrow \mathbb{2Z}$ by $f(a)=a^3+3a+2$, then show that $f$ is not onto. Hint: Show that $f(a)\neq 0$. I have a feeling I have to use the root theorem test, but I ...
2
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4answers
43 views

Domain of the function $f(x) = \sqrt{\frac{3^x-4^x}{x^2-4x-4}}$ will be?

I tried solving this question by $1.$ $-1$ and $4$ will not be in domain because denominator can not be zero . $2.$ Either both denominator and numerator will be positive or negative so that whole ...
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3answers
26 views

Investigating the bijectivity of $ 2 x + |\cos(x)| $.

The question asks if the function $$ f(x) = 2 x + |\cos(x)| $$ if (one-one, onto), (many-one, onto) or (one-one, into). After a long process of plotting the graph, I managed to guess it’s one-one and ...
2
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4answers
55 views

A formal proof that the function $ x \mapsto x^{2} $ is continuous at $ x = 4 $.

Problem: Show $f(x)=x^2 $ is continuous at $ x = 4$. That is to say, find delta such that: $ ∀ε>0$ $ ∃δ>0 $ such that $ |x-a|<δ ⇒ |f(x)-f(a)|<ε$ Where $a=4$, $f(x)=x^2$,and $f(a)=16$. ...
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2answers
14 views

Proof of Injection and Surjection

I am having trouble proving the function f is injective and surjective. $f$ is a function from $\mathbb{Z}\times{Z} \to $\mathbb{Z}\times{Z}$ and $f(x,y) = (5x-y,x+y)$. I know it should be fairly ...
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0answers
21 views

How we get the result of this limit?

I met a problem while doing my homework. Let say we have a formula: $(a_3 + d)\cdot sin(\theta_2) - b_3 \cdot cos(\theta_2) - a_2 = 0$ Now we knew $a_3$, d(in this case is exactly 0), $b_3$ and ...
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1answer
15 views

Functions problem: surjectivity and direct and inverse image theory

I need some help with this problem, if sombody could give me any idea of how to solve it (not the solution itself, but it would be better) I will appreciate it: for a function $f: A → B$, prove $ ∀ Z ...
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1answer
39 views

Prove that $f:\mathbb N\to\mathbb Z$ is a bijection. [on hold]

Define $f:\mathbb N\to\mathbb Z$ by $$ f(n) = \begin{cases} (1-n)/2& \text{if $n$ is odd;}\\ n/2 &\text{if $n$ is even.}\end{cases}$$ Prove that $f$ is a bijection and determine the ...
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votes
1answer
31 views

Prove that α : p(z) -> p(z) is a bijection [on hold]

Please help me prove this: Let $\alpha : \mathscr{P}(\mathbb{Z}) \rightarrow \mathscr{P}(\mathbb{Z})$ be defined by $$\alpha(S)=\begin{cases} S\cup\{0\} \text{ if }0\notin S\\ S\setminus\{0\}\text{ ...
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0answers
25 views

How do I find the domain and range of this piecewise defined function?

Are both conditions true when $x>3$? If so, how do I graph it? $$ f(x)=\left\{\begin{aligned} &x^2-4&&:x>3\\ &2x-1&&:x\geq 3 \end{aligned} \right. $$
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4answers
64 views

if $f:X \to Y$ is 1-1 and $|X| = |Y|$, does that imply $f$ is onto?

Similarly, if $f$ is onto and both sets have the same cardinality, does that imply $f$ is 1-1? I'm pretty sure both statements are true but I'd rather not assume. Thank you for your time.
1
vote
2answers
30 views

Correct to write $\vec{F}:\mathbb{R}^3\rightarrow\mathbb{R}^3$?

Suppose I have some vector field \begin{align} \vec{F}\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)&=G\textbf{i}+H\textbf{j}+T\textbf{k}.\tag{1} \end{align} Would it be correct for ...
1
vote
1answer
40 views

Nice Formula for a Function from $\mathbb{N}\cup\{0\}$

I am trying to get a nice formula for the following function $$f:\mathbb{N}\cup\{0\}\rightarrow \{1,1,-1,3,-3,5,-5,7,-7,9,-9,...\}$$ thus It seems like it would be closely related to somthing like ...
0
votes
1answer
8 views

Is the piece-wise function with mapping $f: (\mathbb{Q^{+}_r} \cup \{0\}) \rightarrow \mathbb{Z^+}$ injective or surjective?

Let $f: (\mathbb{Q^{+}_r} \cup \{0\}) \rightarrow \mathbb{Z^+}$ by $\begin{array}{cc}\Bigg \{&\begin{array}{cc} f(0)=1 \\ f(\frac{a}{b}) = a+b \end{array} \end{array}$ where $\mathbb{Q^{+}_r} = ...
2
votes
2answers
48 views

How to smoothly approximate a sign function

I have a function that defined as following $$f(x) = \begin{cases} 1, & \text{if $x > 0$ } \\ 0, & \text{if $x=0$ } \\ -1, & \text{if $x<0$ } \end{cases}$$ In practice, the $f(x)$ ...