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

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0
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1answer
86 views

Prove that function has no finite limit using $\epsilon$ - $\delta$ defintion

I want to prove using $$(\exists \varepsilon > 0)(\forall \delta > 0)\exists x(0 < \left| {x - {x_0}} \right| < \delta \Rightarrow \left| {f(x) - L} \right| \ge \varepsilon )$$ That the ...
1
vote
1answer
64 views

Find the residue of the function $g(z)=f(z^2)$ at a given point.

Let $f(z)$ be analytic in $0<|z|<R$. Find the residue of the function $g(z)=f(z^2)$ at $z_0=0$. I am looking for a solution to this problem. My thoughts: I know in order to find the residue ...
2
votes
3answers
66 views

Find the inverse of $f(x,y) = (x+3y,3x+y)$

Given the function $f : \mathbb{R}^2 \to \mathbb{R}^2$ as $f(x,y) = (x+3y,3x+y)$. Find $f^{-1}$ .( Assume $f$ is a bijection) I know how to find $f^{-1} (x) = (3x+2)$ or anything with one ...
1
vote
1answer
35 views

Show that if $\lim_{x \to a}f(x)=L$ then $f$ is bounded near $a$.

The Problem: Show that if $\lim_{x \to a}f(x)=L$ then $f$ is bounded near $a$, i.e. there are constants $C,M > 0$ such that $\left|f(x)\right|<M$ for all $x$ such that $\left|x-a\right| < ...
0
votes
2answers
24 views

Function that flows from y=a in a curve to y=x

I'm searching for a function that looks approximately like the red line in this sketch: The function should start horizontally like y=a and flow in a curve to a diagonal line like y=x. I need to give ...
1
vote
1answer
121 views

Proof of functions: onto, one-to-one, and bijection theorem?

Im taking abstract structure of math and am having a hard time with function proofs (set and subset proofs made so much more sense). For class tomorrow we are asked to attempt to prove theorem 3.2 ...
2
votes
4answers
68 views

Let $A= { x_1 , x_2 , x_3 , x_4 ,x_5 }$ , $B = { y_1 , y_2 , y_3 , y_4 , y_5 }$ , then find the number of one-one functions from $A$ to $B$ such that

Let $A= \{ x_1 , x_2 , x_3 , x_4 ,x_5 \}$ , $B = \{ y_1 , y_2 , y_3 , y_4 , y_5 \}$ , then find the number of one-one functions from $A$ to $B$ such that $f(x_i) \ne {y_i}$ where $i = 1,2,3,4,5$ . So ...
1
vote
1answer
44 views

Right notation to recurse over a sequence or list

I have a function $f(x, a)$ which is invoked over all the elements of a sequence feeding the result to the next call, with $x$ being the next element in the list and $a$ the accumulated result. What ...
-5
votes
2answers
48 views

Find the derivative of the function $y=(-6x^{10}+70x-4)^6$ [closed]

Find the derivative of the function $y=(-6x^{10}+70x-4)^6$. Select one $6(-6x^{10}+70x-4)^5$ $-10(-6x^{10}+70x-4)^6(6x^9-7)$ $-10(-6x^{10}+70x-4)^5(6x^9-7)$ ...
0
votes
1answer
19 views

Checking the convexity of a parametric set

Let $r\in\mathbb{R}$ and $|v|\leq \frac{1}{2}$. Prove that $$ \{x\in[0,1]:\sqrt{x}+vx\leq r\} $$ is convex. Thank you for all kind help.
0
votes
1answer
28 views

Power series expansion requirements

Hello stackexchange folks :) I have a question regarding the assumptions made right before you choose to expand or approximate a function by a power series. Specifically I have the function: ...
6
votes
2answers
69 views

Prove if $f'(x)\geq 1$ then $\exists c$ such that $f(c)=0$.

Let $f:\mathbb{R}\to\mathbb{R}$ be differentiable on $\mathbb{R}$ If $f'(x)\geq 1$ for all $x\in \mathbb{R}$, then there exists a $c\in \mathbb{R}$ such that $f(c)=0$. I realised that since ...
1
vote
1answer
225 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 ...
1
vote
1answer
64 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?
-2
votes
2answers
144 views

To prove that no such function can be continuous. [closed]

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.
2
votes
2answers
60 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 ...
0
votes
2answers
68 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
vote
1answer
32 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 $4.$ ...
0
votes
3answers
82 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"? ...
2
votes
1answer
123 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
votes
1answer
102 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 ...
0
votes
2answers
48 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
2answers
85 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 ...
0
votes
1answer
43 views

Prove a function $f$ is one-to-one iff $F^{-1}[f(x)] = x$

Prove a function $f$ is one-to-one iff $F^{-1}[f(x)] = x$. I know that a function is defined to be one-to-one (injective) if given $a$ in the domain such that $f(a)=b$, the inverse image of $b$ ...
1
vote
0answers
40 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 ...
5
votes
2answers
214 views

The composition of a nowhere-differentiable function with a differentiable function.

This is actually Problem $ 17 $ from Chapter $ 10 $ of the Fourth 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
62 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
32 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
31 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
31 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
40 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
42 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 ...
3
votes
2answers
87 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
66 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
3answers
69 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 ...
0
votes
5answers
88 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
45 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 ...
2
votes
1answer
87 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
108 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
27 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
17 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 ...
2
votes
3answers
57 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
194 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
45 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
votes
4answers
57 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
28 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
80 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
23 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
31 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
101 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 ...