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

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0
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2answers
16 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 ...
-3
votes
1answer
18 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 $|A^C|=|B^D|$.
0
votes
0answers
14 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^{ ...
2
votes
2answers
62 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
votes
1answer
28 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 ...
0
votes
0answers
33 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 ...
1
vote
0answers
13 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
votes
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.
1
vote
0answers
21 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 $n$ ...
0
votes
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
votes
1answer
23 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 $k(x)= e^{x^2}$
4
votes
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 ...
0
votes
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
votes
2answers
56 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 $\sin^{-1}(x) = \frac{1}{\sin(x)}$, ...
1
vote
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 ...
-3
votes
1answer
25 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) $p \geq-1/3$
0
votes
4answers
60 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
votes
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
vote
1answer
51 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
vote
2answers
34 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
vote
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 ...
0
votes
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 ...
0
votes
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
votes
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 ...
1
vote
3answers
120 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
votes
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 ...
1
vote
4answers
41 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 ...
2
votes
3answers
25 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
votes
4answers
54 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 $f(a)=16$. ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
-4
votes
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 ...
-3
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{ ...
1
vote
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? $$ f(x)=\left\{\begin{aligned} &x^2-4&&:x>3\\ &2x-1&&:x\geq 3 \end{aligned} \right. $$
0
votes
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
0answers
10 views

How should I create a single score with two values as input?

I have two series of values, a and b as inputs and I want to create a score, c, which reflects both of them equally. The distribution of a and b are below In both cases, the x-axis is just an ...
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 $f(x)$ ...
0
votes
1answer
15 views

Prove that the relation $f$: ($\mathbb{Z^*} \times \mathbb{Z^*}$) $\rightarrow$ $\mathbb{Q}$ by $f(a,b)$ = $\frac{a+b}{a+b-3ab}$ is a function

Prove that the relation $f$: ($\mathbb{Z^*} \times \mathbb{Z^*}$) $\rightarrow$ $\mathbb{Q}$ by $f(a,b)$ = $\frac{a+b}{a+b-3ab}$ is a function I believe that f is a function and I am attempting to ...
-2
votes
2answers
33 views

How do I evaluate these functions [on hold]

(1) $(f\circ g)(x)=3x^2$ and $g(x)=3x^2-1$. What is $f(x)$ (2) $(f\circ g)(x)=9x^2+6x$ and $g(x)=3x+1$, find $f(x)$.
0
votes
0answers
10 views

Formula for a recursive function

Given the recursive function $T: \mathbb{N}_0 \to \mathbb{R}_+$ $$T(n) ≤ max_{1 ≤ k ≤ n - 1}\{T(k-1) + T(n - k) + c n^2\}$$ with c > 0 constant, I want to determine an absolute formula to quickly ...
13
votes
8answers
1k views

Can we have a one-one function from [0,1] to the set of irrational numbers?

Since both of them are uncountable sets, we should be able to construct such a map. Am I correct? If so, then what is the map?
0
votes
2answers
37 views

What is the range of the function $f(x)=\log x+\sin x $?

I've been thinking about this problem for some time now and I initially thought that the range was $\mathbb R$ . I arrived at that conclusion because in a similar problem we had to find the range of ...
1
vote
1answer
12 views

When can you not do a mapping composition?

Suppose I have $\alpha:\mathbb R^3 \to \mathbb R$ and $\beta:\mathbb R \to \mathbb R^+$. Looking over my notes, it says $\alpha \circ \beta$ can not be done but $\beta \circ \alpha$ can. What is the ...
3
votes
2answers
55 views

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$ What I did : Use $g(x) = f(x)-x$, then $g'(x) = f'(x)-1\ne0$ I suspect I have to use Rolle's theorem now, But I am having difficulty ...
9
votes
3answers
299 views

Finding a pair of functions with properties

I need to find a pair of functions $f$, $g$ such that $f$ is not differentiable at $x = 0$ $g$ is not differentiable at $f(0)$ $g \circ f$ is differentiable at $x = 0$ I've tried a lot of ...